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case intro
α✝ : Sort ?u.9034
β : Sort ?u.9037
γ : Sort ?u.9040
f✝ : α✝ → β
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => { fst := U, snd := cast (_ : Set T = f U) s }
hg : Injective g
⊢ False | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
| exact cantor_injective g hg | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
| Mathlib.Logic.Function.Basic.301_0.QX1TCPxnrBJfF8i | /-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | Mathlib_Logic_Function_Basic |
α : Sort u_2
β : Sort u_1
γ : Sort u_3
f✝ f : α → β
g : β → α
h : β → γ
i : γ → β
hf : LeftInverse f g
hh : LeftInverse h i
a : γ
⊢ h (f (g (i a))) = a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by | rw [hf (i a), hh a] | theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by | Mathlib.Logic.Function.Basic.354_0.QX1TCPxnrBJfF8i | theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.11583
f✝ f : α → β
g₁ g₂ : β → α
h₁ : LeftInverse g₁ f
h₂ : RightInverse g₂ f
⊢ g₁ = g₁ ∘ f ∘ g₂ | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by | rw [h₂.comp_eq_id, comp.right_id] | theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by | Mathlib.Logic.Function.Basic.400_0.QX1TCPxnrBJfF8i | theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort ?u.11583
f✝ f : α → β
g₁ g₂ : β → α
h₁ : LeftInverse g₁ f
h₂ : RightInverse g₂ f
⊢ g₁ ∘ f ∘ g₂ = g₂ | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by | rw [← comp.assoc, h₁.comp_eq_id, comp.left_id] | theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by | Mathlib.Logic.Function.Basic.400_0.QX1TCPxnrBJfF8i | theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
a : α
b : β
h : partialInv f b = some a
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ∃ a, f a = b
⊢ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by | rw [hpi, dif_pos h'] at h | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by | Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
a : α
b : β
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ∃ a, f a = b
h : some (Classical.choose h') = some a
⊢ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
| injection h with h | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
| Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
a : α
b : β
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ∃ a, f a = b
h : Classical.choose h' = a
⊢ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
| subst h | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
| Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
b : β
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ∃ a, f a = b
⊢ f (Classical.choose h') = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
| apply Classical.choose_spec h' | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
| Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
a : α
b : β
h : partialInv f b = some a
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ¬∃ a, f a = b
⊢ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by | rw [hpi, dif_neg h'] at h | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by | Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α✝ : Sort ?u.12049
β✝ : Sort ?u.12052
γ : Sort ?u.12055
f✝ : α✝ → β✝
α : Type u_1
β : Sort u_2
f : α → β
I : Injective f
a : α
b : β
h : none = some a
hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none
h' : ¬∃ a, f a = b
⊢ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; | contradiction | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; | Mathlib.Logic.Function.Basic.415_0.QX1TCPxnrBJfF8i | theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
inst✝ : Nonempty α
f : α → β
a : α
b : β
h : ∃ a, f a = b
⊢ f (invFun f b) = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by | simp only [invFun, dif_pos h, h.choose_spec] | theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by | Mathlib.Logic.Function.Basic.449_0.QX1TCPxnrBJfF8i | theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
inst✝ : Nonempty α
f : α → β
a : α
b✝ : β
g : β → α
hf : Injective f
hg : RightInverse g f
b : β
⊢ f (invFun f b) = f (g b) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
| rw [hg b] | theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
| Mathlib.Logic.Function.Basic.461_0.QX1TCPxnrBJfF8i | theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
inst✝ : Nonempty α
f : α → β
a : α
b✝ : β
g : β → α
hf : Injective f
hg : RightInverse g f
b : β
⊢ f (invFun f b) = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
| exact invFun_eq ⟨g b, hg b⟩ | theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
| Mathlib.Logic.Function.Basic.461_0.QX1TCPxnrBJfF8i | theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g | Mathlib_Logic_Function_Basic |
α : Sort u
β✝ : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β✝ a
a✝ : α
b✝ : β✝ a✝
β : Sort u_1
f : α → β
a' : α
b : β
a : α
⊢ update f a' b a = if a = a' then b else f a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
| rcases Decidable.eq_or_ne a a' with rfl | hne | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
| Mathlib.Logic.Function.Basic.575_0.QX1TCPxnrBJfF8i | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a | Mathlib_Logic_Function_Basic |
case inl
α : Sort u
β✝ : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β✝ a
a✝ : α
b✝ : β✝ a✝
β : Sort u_1
f : α → β
b : β
a : α
⊢ update f a b a = if a = a then b else f a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> | simp [*] | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> | Mathlib.Logic.Function.Basic.575_0.QX1TCPxnrBJfF8i | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a | Mathlib_Logic_Function_Basic |
case inr
α : Sort u
β✝ : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β✝ a
a✝ : α
b✝ : β✝ a✝
β : Sort u_1
f : α → β
a' : α
b : β
a : α
hne : a ≠ a'
⊢ update f a' b a = if a = a' then b else f a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> | simp [*] | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> | Mathlib.Logic.Function.Basic.575_0.QX1TCPxnrBJfF8i | /-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a : α
b : β a
f : (a : α) → β a
a' : α
v v' : β a'
h : update f a' v = update f a' v'
⊢ v = v' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
| have := congr_fun h a' | theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
| Mathlib.Logic.Function.Basic.592_0.QX1TCPxnrBJfF8i | theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a : α
b : β a
f : (a : α) → β a
a' : α
v v' : β a'
h : update f a' v = update f a' v'
this : update f a' v a' = update f a' v' a'
⊢ v = v' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
| rwa [update_same, update_same] at this | theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
| Mathlib.Logic.Function.Basic.592_0.QX1TCPxnrBJfF8i | theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a✝ : α
b✝ : β a✝
f : (a : α) → β a
a : α
b : β a
p : (a : α) → β a → Prop
⊢ (∀ (x : α), p x (update f a b x)) ↔ p a b ∧ ∀ (x : α), x ≠ a → p x (f x) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
| rw [← and_forall_ne a, update_same] | lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
| Mathlib.Logic.Function.Basic.597_0.QX1TCPxnrBJfF8i | lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a✝ : α
b✝ : β a✝
f : (a : α) → β a
a : α
b : β a
p : (a : α) → β a → Prop
⊢ (p a b ∧ ∀ (b_1 : α), b_1 ≠ a → p b_1 (update f a b b_1)) ↔ p a b ∧ ∀ (x : α), x ≠ a → p x (f x) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
| simp (config := { contextual := true }) | lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
| Mathlib.Logic.Function.Basic.597_0.QX1TCPxnrBJfF8i | lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a✝ : α
b✝ : β a✝
f : (a : α) → β a
a : α
b : β a
p : (a : α) → β a → Prop
⊢ (∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ x x_1, p x (f x) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
| rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b] | theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
| Mathlib.Logic.Function.Basic.603_0.QX1TCPxnrBJfF8i | theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a✝ : α
b✝ : β a✝
f : (a : α) → β a
a : α
b : β a
p : (a : α) → β a → Prop
⊢ ¬(¬p a b ∧ ∀ (x : α), x ≠ a → ¬p x (f x)) ↔ p a b ∨ ∃ x x_1, p x (f x) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
| simp [-not_and, not_and_or] | theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
| Mathlib.Logic.Function.Basic.603_0.QX1TCPxnrBJfF8i | theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f g : (a : α) → β a
a : α
b : β a
⊢ update f a b = f ↔ b = f a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by | simp [update_eq_iff] | @[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by | Mathlib.Logic.Function.Basic.619_0.QX1TCPxnrBJfF8i | @[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f g : (a : α) → β a
a : α
b : β a
⊢ f = update f a b ↔ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by | simp [eq_update_iff] | @[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by | Mathlib.Logic.Function.Basic.622_0.QX1TCPxnrBJfF8i | @[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b | Mathlib_Logic_Function_Basic |
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
f : (i : ι) → α i → β i
g : (i : ι) → α i
i : ι
v : α i
j : ι
⊢ f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
| by_cases h:j = i | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
| Mathlib.Logic.Function.Basic.659_0.QX1TCPxnrBJfF8i | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
f : (i : ι) → α i → β i
g : (i : ι) → α i
i : ι
v : α i
j : ι
h : j = i
⊢ f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· | subst j | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· | Mathlib.Logic.Function.Basic.659_0.QX1TCPxnrBJfF8i | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
f : (i : ι) → α i → β i
g : (i : ι) → α i
i : ι
v : α i
⊢ f i (update g i v i) = update (fun k => f k (g k)) i (f i v) i | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
| simp | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
| Mathlib.Logic.Function.Basic.659_0.QX1TCPxnrBJfF8i | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
f : (i : ι) → α i → β i
g : (i : ι) → α i
i : ι
v : α i
j : ι
h : ¬j = i
⊢ f j (update g i v j) = update (fun k => f k (g k)) i (f i v) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· | simp [h] | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· | Mathlib.Logic.Function.Basic.659_0.QX1TCPxnrBJfF8i | theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j | Mathlib_Logic_Function_Basic |
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
γ : ι → Sort u_4
f : (i : ι) → α i → β i → γ i
g : (i : ι) → α i
h : (i : ι) → β i
i : ι
v : α i
w : β i
j : ι
⊢ f j (update g i v j) (update h i w j) = update (fun k => f k (g k) (h k)) i (f i v w) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
| by_cases h:j = i | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
| Mathlib.Logic.Function.Basic.668_0.QX1TCPxnrBJfF8i | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
γ : ι → Sort u_4
f : (i : ι) → α i → β i → γ i
g : (i : ι) → α i
h✝ : (i : ι) → β i
i : ι
v : α i
w : β i
j : ι
h : j = i
⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· | subst j | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· | Mathlib.Logic.Function.Basic.668_0.QX1TCPxnrBJfF8i | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
γ : ι → Sort u_4
f : (i : ι) → α i → β i → γ i
g : (i : ι) → α i
h : (i : ι) → β i
i : ι
v : α i
w : β i
⊢ f i (update g i v i) (update h i w i) = update (fun k => f k (g k) (h k)) i (f i v w) i | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
| simp | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
| Mathlib.Logic.Function.Basic.668_0.QX1TCPxnrBJfF8i | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g✝ : (a : α✝) → β✝ a
a : α✝
b : β✝ a
ι : Sort u_1
inst✝ : DecidableEq ι
α : ι → Sort u_2
β : ι → Sort u_3
γ : ι → Sort u_4
f : (i : ι) → α i → β i → γ i
g : (i : ι) → α i
h✝ : (i : ι) → β i
i : ι
v : α i
w : β i
j : ι
h : ¬j = i
⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· | simp [h] | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· | Mathlib.Logic.Function.Basic.668_0.QX1TCPxnrBJfF8i | theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j | Mathlib_Logic_Function_Basic |
α : Sort u
β : α → Sort v
α' : Sort w
inst✝¹ : DecidableEq α
inst✝ : DecidableEq α'
f✝ g : (a : α) → β a
a✝ : α
b : β a✝
P : ⦃a : α⦄ → β a → Prop
f : (a : α) → β a
a' : α
v : β a'
a : α
⊢ P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
| rw [apply_update P, update_apply, ite_prop_iff_or] | theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
| Mathlib.Logic.Function.Basic.677_0.QX1TCPxnrBJfF8i | theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) | Mathlib_Logic_Function_Basic |
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
⊢ update (update f a v) b w = update (update f b w) a v | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
| funext c | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
| Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case h
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
⊢ update (update f a v) b w c = update (update f b w) a v c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
| simp only [update] | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
| Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case h
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
| by_cases h₁ : c = b | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
| Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : c = b
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> | by_cases h₂ : c = a | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : ¬c = b
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> | by_cases h₂ : c = a | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : c = b
h₂ : c = a
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· | rw [dif_pos h₁, dif_pos h₂] | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : c = b
h₂ : c = a
⊢ (_ : b = c) ▸ w = (_ : a = c) ▸ v | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
| cases h (h₂.symm.trans h₁) | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
| Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : c = b
h₂ : ¬c = a
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· | rw [dif_pos h₁, dif_pos h₁, dif_neg h₂] | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : ¬c = b
h₂ : c = a
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· | rw [dif_neg h₁, dif_neg h₁, dif_pos h₂] | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a b : α
h : a ≠ b
v : β a
w : β b
f : (a : α) → β a
c : α
h₁ : ¬c = b
h₂ : ¬c = a
⊢ (if h : c = b then (_ : b = c) ▸ w else if h : c = a then (_ : a = c) ▸ v else f c) =
if h : c = a then (_ : a = c) ▸ v else if h : c = b then (_ : b = c) ▸ w else f c | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· | rw [dif_neg h₁, dif_neg h₁, dif_neg h₂] | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· | Mathlib.Logic.Function.Basic.686_0.QX1TCPxnrBJfF8i | theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v | Mathlib_Logic_Function_Basic |
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a : α
v w : β a
f : (a : α) → β a
⊢ update (update f a v) a w = update f a w | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
| funext b | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
| Mathlib.Logic.Function.Basic.698_0.QX1TCPxnrBJfF8i | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w | Mathlib_Logic_Function_Basic |
case h
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a : α
v w : β a
f : (a : α) → β a
b : α
⊢ update (update f a v) a w b = update f a w b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
| by_cases h : b = a | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
| Mathlib.Logic.Function.Basic.698_0.QX1TCPxnrBJfF8i | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w | Mathlib_Logic_Function_Basic |
case pos
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a : α
v w : β a
f : (a : α) → β a
b : α
h : b = a
⊢ update (update f a v) a w b = update f a w b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> | simp [update, h] | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> | Mathlib.Logic.Function.Basic.698_0.QX1TCPxnrBJfF8i | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w | Mathlib_Logic_Function_Basic |
case neg
α✝ : Sort u
β✝ : α✝ → Sort v
α' : Sort w
inst✝² : DecidableEq α✝
inst✝¹ : DecidableEq α'
f✝ g : (a : α✝) → β✝ a
a✝ : α✝
b✝ : β✝ a✝
α : Sort u_2
inst✝ : DecidableEq α
β : α → Sort u_1
a : α
v w : β a
f : (a : α) → β a
b : α
h : ¬b = a
⊢ update (update f a v) a w b = update f a w b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> | simp [update, h] | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> | Mathlib.Logic.Function.Basic.698_0.QX1TCPxnrBJfF8i | @[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f✝ f : α → β
g : α → γ
e' : β → γ
b : β
inst✝ : Decidable (∃ a, f a = b)
⊢ extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
| unfold extend | theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
| Mathlib.Logic.Function.Basic.734_0.QX1TCPxnrBJfF8i | theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f✝ f : α → β
g : α → γ
e' : β → γ
b : β
inst✝ : Decidable (∃ a, f a = b)
⊢ (if h : ∃ a, f a = b then g (Classical.choose h) else e' b) =
if h : ∃ a, f a = b then g (Classical.choose h) else e' b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
| congr | theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
| Mathlib.Logic.Function.Basic.734_0.QX1TCPxnrBJfF8i | theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
hf : FactorsThrough g f
e' : β → γ
a : α
⊢ extend f g e' (f a) = g a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
| simp only [extend_def, dif_pos, exists_apply_eq_apply] | lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
| Mathlib.Logic.Function.Basic.744_0.QX1TCPxnrBJfF8i | lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
hf : FactorsThrough g f
e' : β → γ
a : α
⊢ g (Classical.choose (_ : ∃ a_1, f a_1 = f a)) = g a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
| exact hf (Classical.choose_spec (exists_apply_eq_apply f a)) | lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
| Mathlib.Logic.Function.Basic.744_0.QX1TCPxnrBJfF8i | lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
e' : β → γ
b : β
hb : ¬∃ a, f a = b
⊢ extend f g e' b = e' b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
| simp [Function.extend_def, hb] | @[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
| Mathlib.Logic.Function.Basic.756_0.QX1TCPxnrBJfF8i | @[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
inst✝ : Nonempty γ
hf : FactorsThrough g f
x : α
⊢ g x = (extend f g (const β (Classical.arbitrary γ)) ∘ f) x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by | simp only [comp_apply, hf.extend_apply] | lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by | Mathlib.Logic.Function.Basic.762_0.QX1TCPxnrBJfF8i | lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
g : α → γ
inst✝ : Nonempty γ
h : ∃ e, g = e ∘ f
x✝¹ x✝ : α
hf : f x✝¹ = f x✝
⊢ g x✝¹ = g x✝ | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by | rw [Classical.choose_spec h, comp_apply, comp_apply, hf] | lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by | Mathlib.Logic.Function.Basic.762_0.QX1TCPxnrBJfF8i | lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Injective f
e' : β → γ
⊢ Injective fun g => extend f g e' | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
| intro g₁ g₂ hg | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
| Mathlib.Logic.Function.Basic.774_0.QX1TCPxnrBJfF8i | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Injective f
e' : β → γ
g₁ g₂ : α → γ
hg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂
⊢ g₁ = g₂ | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
| refine' funext fun x ↦ _ | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
| Mathlib.Logic.Function.Basic.774_0.QX1TCPxnrBJfF8i | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Injective f
e' : β → γ
g₁ g₂ : α → γ
hg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂
x : α
⊢ g₁ x = g₂ x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
| have H := congr_fun hg (f x) | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
| Mathlib.Logic.Function.Basic.774_0.QX1TCPxnrBJfF8i | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Injective f
e' : β → γ
g₁ g₂ : α → γ
hg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂
x : α
H : (fun g => extend f g e') g₁ (f x) = (fun g => extend f g e') g₂ (f x)
⊢ g₁ x = g₂ x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
| simp only [hf.extend_apply] at H | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
| Mathlib.Logic.Function.Basic.774_0.QX1TCPxnrBJfF8i | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Injective f
e' : β → γ
g₁ g₂ : α → γ
hg : (fun g => extend f g e') g₁ = (fun g => extend f g e') g₂
x : α
H : g₁ x = g₂ x
⊢ g₁ x = g₂ x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
| exact H | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
| Mathlib.Logic.Function.Basic.774_0.QX1TCPxnrBJfF8i | theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : Sort u_3
f : α → β
hf : Bijective f
g : α → γ
⊢ (fun g => g ∘ f) (g ∘ surjInv (_ : Surjective f)) = g | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by | simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id] | theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by | Mathlib.Logic.Function.Basic.802_0.QX1TCPxnrBJfF8i | theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f | Mathlib_Logic_Function_Basic |
α : Sort u
f : α → α
h : Involutive f
P : Prop
inst✝ : Decidable P
x : α
⊢ f (if P then x else f x) = if ¬P then x else f x | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by | rw [apply_ite f, h, ite_not] | /-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by | Mathlib.Logic.Function.Basic.909_0.QX1TCPxnrBJfF8i | /-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) | Mathlib_Logic_Function_Basic |
α : Sort u_1
f : α → α
⊢ (Symmetric fun x x_1 => f x = x_1) ↔ Involutive f | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
| simp [Symmetric, Involutive] | @[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
| Mathlib.Logic.Function.Basic.921_0.QX1TCPxnrBJfF8i | @[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f | Mathlib_Logic_Function_Basic |
p : Prop
α : Sort u_1
inst✝ : Nonempty α
P : α → Prop
f : p → α
a : p
h : P (f a)
⊢ P (sometimes f) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by | rwa [sometimes_eq] | theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by | Mathlib.Logic.Function.Basic.985_0.QX1TCPxnrBJfF8i | theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
r : α → β → Prop
⊢ (∀ (a : α), ∃! b, r a b) ↔ ∃ f, ∀ {a : α} {b : β}, r a b ↔ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
| refine ⟨fun h ↦ ?_, ?_⟩ | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
| Mathlib.Logic.Function.Basic.994_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
case refine_1
α : Sort u_1
β : Sort u_2
r : α → β → Prop
h : ∀ (a : α), ∃! b, r a b
⊢ ∃ f, ∀ {a : α} {b : β}, r a b ↔ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· | refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩ | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· | Mathlib.Logic.Function.Basic.994_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
case refine_1.refine_1
α : Sort u_1
β : Sort u_2
r : α → β → Prop
h : ∀ (a : α), ∃! b, r a b
a✝ : α
b✝ : β
hr : r a✝ b✝
⊢ (fun a => Exists.choose (_ : ∃! b, r a b)) a✝ = b✝
case refine_1.refine_2
α : Sort u_1
β : Sort u_2
r : α → β → Prop
h : ∀ (a : α), ∃! b, r a b
a✝ : α
b✝ : β
h' : (fun a => Exists.choose (_ : ∃! b, r a b)) a✝ = b✝
⊢ r a✝ ((fun a => Exists.choose (_ : ∃! b, r a b)) a✝) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
| exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1] | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
| Mathlib.Logic.Function.Basic.994_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
case refine_2
α : Sort u_1
β : Sort u_2
r : α → β → Prop
⊢ (∃ f, ∀ {a : α} {b : β}, r a b ↔ f a = b) → ∀ (a : α), ∃! b, r a b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· | rintro ⟨f, hf⟩ | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· | Mathlib.Logic.Function.Basic.994_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
case refine_2.intro
α : Sort u_1
β : Sort u_2
r : α → β → Prop
f : α → β
hf : ∀ {a : α} {b : β}, r a b ↔ f a = b
⊢ ∀ (a : α), ∃! b, r a b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
| simp [hf] | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
| Mathlib.Logic.Function.Basic.994_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
r : α → β → Prop
⊢ (∀ (a : α), ∃! b, r a b) ↔ ∃ f, r = fun x x_1 => f x = x_1 | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
| simp [forall_existsUnique_iff, Function.funext_iff] | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
| Mathlib.Logic.Function.Basic.1005_0.QX1TCPxnrBJfF8i | /-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) | Mathlib_Logic_Function_Basic |
α : Sort u_1
r : α → α → Prop
hr : Symmetric r
⊢ (∀ (a : α), ∃! b, r a b) ↔ ∃ f, Involutive f ∧ r = fun x x_1 => f x = x_1 | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
| refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩ | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
| Mathlib.Logic.Function.Basic.1012_0.QX1TCPxnrBJfF8i | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) | Mathlib_Logic_Function_Basic |
α : Sort u_1
r : α → α → Prop
hr : Symmetric r
h : ∀ (a : α), ∃! b, r a b
⊢ ∃ f, Involutive f ∧ r = fun x x_1 => f x = x_1 | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
| rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩ | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
| Mathlib.Logic.Function.Basic.1012_0.QX1TCPxnrBJfF8i | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) | Mathlib_Logic_Function_Basic |
case intro
α : Sort u_1
f : α → α
hr : Symmetric fun x x_1 => f x = x_1
h : ∀ (a : α), ∃! b, (fun x x_1 => f x = x_1) a b
⊢ ∃ f_1, Involutive f_1 ∧ (fun x x_1 => f x = x_1) = fun x x_1 => f_1 x = x_1 | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
| exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩ | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
| Mathlib.Logic.Function.Basic.1012_0.QX1TCPxnrBJfF8i | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) | Mathlib_Logic_Function_Basic |
α : Sort u_1
r : α → α → Prop
hr : Symmetric r
⊢ (∀ (a : α), ∃! b, r a b) ↔ ∃ f, Involutive f ∧ ∀ {a b : α}, r a b ↔ f a = b | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
| simp [hr.forall_existsUnique_iff', funext_iff] | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
| Mathlib.Logic.Function.Basic.1021_0.QX1TCPxnrBJfF8i | /-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b | Mathlib_Logic_Function_Basic |
α β : Sort u_1
h : α = β
⊢ Bijective (Eq.mp h) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
| cases h | theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
| Mathlib.Logic.Function.Basic.1042_0.QX1TCPxnrBJfF8i | theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) | Mathlib_Logic_Function_Basic |
case refl
α : Sort u_1
⊢ Bijective (Eq.mp (_ : α = α)) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
| refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩ | theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
| Mathlib.Logic.Function.Basic.1042_0.QX1TCPxnrBJfF8i | theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) | Mathlib_Logic_Function_Basic |
α β : Sort u_1
h : α = β
⊢ Bijective (Eq.mpr h) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
| cases h | theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
| Mathlib.Logic.Function.Basic.1049_0.QX1TCPxnrBJfF8i | theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) | Mathlib_Logic_Function_Basic |
case refl
α : Sort u_1
⊢ Bijective (Eq.mpr (_ : α = α)) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
| refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩ | theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
| Mathlib.Logic.Function.Basic.1049_0.QX1TCPxnrBJfF8i | theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) | Mathlib_Logic_Function_Basic |
α β : Sort u_1
h : α = β
⊢ Bijective (cast h) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mpr_bijective eq_mpr_bijective
theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
| cases h | theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
| Mathlib.Logic.Function.Basic.1054_0.QX1TCPxnrBJfF8i | theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) | Mathlib_Logic_Function_Basic |
case refl
α : Sort u_1
⊢ Bijective (cast (_ : α = α)) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mpr_bijective eq_mpr_bijective
theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
cases h
| refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩ | theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
cases h
| Mathlib.Logic.Function.Basic.1054_0.QX1TCPxnrBJfF8i | theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : β → Sort v
f : α → β
g : β → α
h : LeftInverse g f
C : (a : α) → γ (f a)
a : α
⊢ HEq (C (g (f a))) (C a) | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mpr_bijective eq_mpr_bijective
theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align cast_bijective cast_bijective
/-! Note these lemmas apply to `Type*` not `Sort*`, as the latter interferes with `simp`, and
is trivial anyway.-/
@[simp]
theorem eq_rec_inj {α : Sort*} {a a' : α} (h : a = a') {C : α → Type*} (x y : C a) :
(Eq.ndrec x h : C a') = Eq.ndrec y h ↔ x = y :=
(eq_rec_on_bijective h).injective.eq_iff
#align eq_rec_inj eq_rec_inj
@[simp]
theorem cast_inj {α β : Type u} (h : α = β) {x y : α} : cast h x = cast h y ↔ x = y :=
(cast_bijective h).injective.eq_iff
#align cast_inj cast_inj
theorem Function.LeftInverse.eq_rec_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
@Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a :=
eq_of_heq <| (eq_rec_heq _ _).trans <| by | rw [h] | theorem Function.LeftInverse.eq_rec_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
@Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a :=
eq_of_heq <| (eq_rec_heq _ _).trans <| by | Mathlib.Logic.Function.Basic.1074_0.QX1TCPxnrBJfF8i | theorem Function.LeftInverse.eq_rec_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
@Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a | Mathlib_Logic_Function_Basic |
α : Sort u_1
β : Sort u_2
γ : β → Sort v
f : α → β
g : β → α
h : LeftInverse g f
C : (a : α) → γ (f a)
a : α
⊢ cast (_ : γ (f (g (f a))) = γ (f a)) (C (g (f a))) = C a | /-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
/-!
# Miscellaneous function constructions and lemmas
-/
set_option autoImplicit true
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort _} {f : α → β}
/-- Evaluate a function at an argument. Useful if you want to talk about the partially applied
`Function.eval x : (∀ x, β x) → β x`. -/
@[reducible, simp] def eval {β : α → Sort*} (x : α) (f : ∀ x, β x) : β x := f x
#align function.eval Function.eval
theorem eval_apply {β : α → Sort*} (x : α) (f : ∀ x, β x) : eval x f = f x :=
rfl
#align function.eval_apply Function.eval_apply
theorem const_def {y : β} : (fun _ : α ↦ y) = const α y :=
rfl
#align function.const_def Function.const_def
@[simp]
theorem const_comp {f : α → β} {c : γ} : const β c ∘ f = const α c :=
rfl
#align function.const_comp Function.const_comp
@[simp]
theorem comp_const {f : β → γ} {b : β} : f ∘ const α b = const α (f b) :=
rfl
#align function.comp_const Function.comp_const
theorem const_injective [Nonempty α] : Injective (const α : β → α → β) := fun y₁ y₂ h ↦
let ⟨x⟩ := ‹Nonempty α›
congr_fun h x
#align function.const_injective Function.const_injective
@[simp]
theorem const_inj [Nonempty α] {y₁ y₂ : β} : const α y₁ = const α y₂ ↔ y₁ = y₂ :=
⟨fun h ↦ const_injective h, fun h ↦ h ▸ rfl⟩
#align function.const_inj Function.const_inj
theorem id_def : @id α = fun x ↦ x :=
rfl
#align function.id_def Function.id_def
-- porting note: `Function.onFun` is now reducible
-- @[simp]
theorem onFun_apply (f : β → β → γ) (g : α → β) (a b : α) : onFun f g a b = f (g a) (g b) :=
rfl
#align function.on_fun_apply Function.onFun_apply
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f' := by
subst hα
have : ∀a, HEq (f a) (f' a) := λ a => h a a (HEq.refl a)
have : β = β' := by funext a
exact type_eq_of_heq (this a)
subst this
apply heq_of_eq
funext a
exact eq_of_heq (this a)
#align function.hfunext Function.hfunext
theorem funext_iff {β : α → Sort*} {f₁ f₂ : ∀ x : α, β x} : f₁ = f₂ ↔ ∀ a, f₁ a = f₂ a :=
Iff.intro (fun h _ ↦ h ▸ rfl) funext
#align function.funext_iff Function.funext_iff
theorem ne_iff {β : α → Sort*} {f₁ f₂ : ∀ a, β a} : f₁ ≠ f₂ ↔ ∃ a, f₁ a ≠ f₂ a :=
funext_iff.not.trans not_forall
#align function.ne_iff Function.ne_iff
protected theorem Bijective.injective {f : α → β} (hf : Bijective f) : Injective f := hf.1
#align function.bijective.injective Function.Bijective.injective
protected theorem Bijective.surjective {f : α → β} (hf : Bijective f) : Surjective f := hf.2
#align function.bijective.surjective Function.Bijective.surjective
theorem Injective.eq_iff (I : Injective f) {a b : α} : f a = f b ↔ a = b :=
⟨@I _ _, congr_arg f⟩
#align function.injective.eq_iff Function.Injective.eq_iff
theorem Injective.beq_eq [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β]
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
theorem Injective.eq_iff' (I : Injective f) {a b : α} {c : β} (h : f b = c) : f a = c ↔ a = b :=
h ▸ I.eq_iff
#align function.injective.eq_iff' Function.Injective.eq_iff'
theorem Injective.ne (hf : Injective f) {a₁ a₂ : α} : a₁ ≠ a₂ → f a₁ ≠ f a₂ :=
mt fun h ↦ hf h
#align function.injective.ne Function.Injective.ne
theorem Injective.ne_iff (hf : Injective f) {x y : α} : f x ≠ f y ↔ x ≠ y :=
⟨mt <| congr_arg f, hf.ne⟩
#align function.injective.ne_iff Function.Injective.ne_iff
theorem Injective.ne_iff' (hf : Injective f) {x y : α} {z : β} (h : f y = z) : f x ≠ z ↔ x ≠ y :=
h ▸ hf.ne_iff
#align function.injective.ne_iff' Function.Injective.ne_iff'
/-- If the co-domain `β` of an injective function `f : α → β` has decidable equality, then
the domain `α` also has decidable equality. -/
protected def Injective.decidableEq [DecidableEq β] (I : Injective f) : DecidableEq α :=
fun _ _ ↦ decidable_of_iff _ I.eq_iff
#align function.injective.decidable_eq Function.Injective.decidableEq
theorem Injective.of_comp {g : γ → α} (I : Injective (f ∘ g)) : Injective g :=
fun _ _ h ↦ I <| congr_arg f h
#align function.injective.of_comp Function.Injective.of_comp
@[simp]
theorem Injective.of_comp_iff (hf : Injective f) (g : γ → α) :
Injective (f ∘ g) ↔ Injective g :=
⟨Injective.of_comp, hf.comp⟩
#align function.injective.of_comp_iff Function.Injective.of_comp_iff
theorem Injective.of_comp_right {g : γ → α} (I : Injective (f ∘ g)) (hg : Surjective g) :
Injective f := fun x y h ↦ by
obtain ⟨x, rfl⟩ := hg x
obtain ⟨y, rfl⟩ := hg y
exact congr_arg g (I h)
theorem Surjective.bijective₂_of_injective {g : γ → α} (hf : Surjective f) (hg : Surjective g)
(I : Injective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨I.of_comp_right hg, hf⟩, I.of_comp, hg⟩
@[simp]
theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g) :
Injective (f ∘ g) ↔ Injective f :=
⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
#align function.injective.of_comp_iff' Function.Injective.of_comp_iff'
/-- Composition by an injective function on the left is itself injective. -/
theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
Function.Injective (g ∘ · : (α → β) → α → γ) :=
fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
#align function.injective.comp_left Function.Injective.comp_left
theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
fun _ _ _ ↦ Subsingleton.elim _ _
#align function.injective_of_subsingleton Function.injective_of_subsingleton
lemma Injective.dite (p : α → Prop) [DecidablePred p]
{f : {a : α // p a} → β} {f' : {a : α // ¬ p a} → β}
(hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (λ x => if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) :=
by intros x₁ x₂ h
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
· rw [dif_pos h₁, dif_pos h₂] at h; injection (hf h)
· rw [dif_pos h₁, dif_neg h₂] at h; exact (im_disj h).elim
· rw [dif_neg h₁, dif_pos h₂] at h; exact (im_disj h.symm).elim
· rw [dif_neg h₁, dif_neg h₂] at h; injection (hf' h)
#align function.injective.dite Function.Injective.dite
theorem Surjective.of_comp {g : γ → α} (S : Surjective (f ∘ g)) : Surjective f := fun y ↦
let ⟨x, h⟩ := S y
⟨g x, h⟩
#align function.surjective.of_comp Function.Surjective.of_comp
@[simp]
theorem Surjective.of_comp_iff (f : α → β) {g : γ → α} (hg : Surjective g) :
Surjective (f ∘ g) ↔ Surjective f :=
⟨Surjective.of_comp, fun h ↦ h.comp hg⟩
#align function.surjective.of_comp_iff Function.Surjective.of_comp_iff
theorem Surjective.of_comp_left {g : γ → α} (S : Surjective (f ∘ g)) (hf : Injective f) :
Surjective g := fun a ↦ let ⟨c, hc⟩ := S (f a); ⟨c, hf hc⟩
theorem Injective.bijective₂_of_surjective {g : γ → α} (hf : Injective f) (hg : Injective g)
(S : Surjective (f ∘ g)) : Bijective f ∧ Bijective g :=
⟨⟨hf, S.of_comp⟩, hg, S.of_comp_left hf⟩
@[simp]
theorem Surjective.of_comp_iff' (hf : Bijective f) (g : γ → α) :
Surjective (f ∘ g) ↔ Surjective g :=
⟨fun S ↦ S.of_comp_left hf.1, hf.surjective.comp⟩
#align function.surjective.of_comp_iff' Function.Surjective.of_comp_iff'
instance decidableEqPFun (p : Prop) [Decidable p] (α : p → Type*) [∀ hp, DecidableEq (α hp)] :
DecidableEq (∀ hp, α hp)
| f, g => decidable_of_iff (∀ hp, f hp = g hp) funext_iff.symm
protected theorem Surjective.forall (hf : Surjective f) {p : β → Prop} :
(∀ y, p y) ↔ ∀ x, p (f x) :=
⟨fun h x ↦ h (f x), fun h y ↦
let ⟨x, hx⟩ := hf y
hx ▸ h x⟩
#align function.surjective.forall Function.Surjective.forall
protected theorem Surjective.forall₂ (hf : Surjective f) {p : β → β → Prop} :
(∀ y₁ y₂, p y₁ y₂) ↔ ∀ x₁ x₂, p (f x₁) (f x₂) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall
#align function.surjective.forall₂ Function.Surjective.forall₂
protected theorem Surjective.forall₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∀ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∀ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.forall.trans $ forall_congr' fun _ ↦ hf.forall₂
#align function.surjective.forall₃ Function.Surjective.forall₃
protected theorem Surjective.exists (hf : Surjective f) {p : β → Prop} :
(∃ y, p y) ↔ ∃ x, p (f x) :=
⟨fun ⟨y, hy⟩ ↦
let ⟨x, hx⟩ := hf y
⟨x, hx.symm ▸ hy⟩,
fun ⟨x, hx⟩ ↦ ⟨f x, hx⟩⟩
#align function.surjective.exists Function.Surjective.exists
protected theorem Surjective.exists₂ (hf : Surjective f) {p : β → β → Prop} :
(∃ y₁ y₂, p y₁ y₂) ↔ ∃ x₁ x₂, p (f x₁) (f x₂) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists
#align function.surjective.exists₂ Function.Surjective.exists₂
protected theorem Surjective.exists₃ (hf : Surjective f) {p : β → β → β → Prop} :
(∃ y₁ y₂ y₃, p y₁ y₂ y₃) ↔ ∃ x₁ x₂ x₃, p (f x₁) (f x₂) (f x₃) :=
hf.exists.trans <| exists_congr fun _ ↦ hf.exists₂
#align function.surjective.exists₃ Function.Surjective.exists₃
theorem Surjective.injective_comp_right (hf : Surjective f) : Injective fun g : β → γ ↦ g ∘ f :=
fun _ _ h ↦ funext <| hf.forall.2 <| congr_fun h
#align function.surjective.injective_comp_right Function.Surjective.injective_comp_right
protected theorem Surjective.right_cancellable (hf : Surjective f) {g₁ g₂ : β → γ} :
g₁ ∘ f = g₂ ∘ f ↔ g₁ = g₂ :=
hf.injective_comp_right.eq_iff
#align function.surjective.right_cancellable Function.Surjective.right_cancellable
theorem surjective_of_right_cancellable_Prop (h : ∀ g₁ g₂ : β → Prop, g₁ ∘ f = g₂ ∘ f → g₁ = g₂) :
Surjective f := by
specialize h (fun y ↦ ∃ x, f x = y) (fun _ ↦ True) (funext fun x ↦ eq_true ⟨_, rfl⟩)
intro y; rw [congr_fun h y]; trivial
#align function.surjective_of_right_cancellable_Prop Function.surjective_of_right_cancellable_Prop
theorem bijective_iff_existsUnique (f : α → β) : Bijective f ↔ ∀ b : β, ∃! a : α, f a = b :=
⟨fun hf b ↦
let ⟨a, ha⟩ := hf.surjective b
⟨a, ha, fun _ ha' ↦ hf.injective (ha'.trans ha.symm)⟩,
fun he ↦ ⟨fun {_a a'} h ↦ (he (f a')).unique h rfl, fun b ↦ (he b).exists⟩⟩
#align function.bijective_iff_exists_unique Function.bijective_iff_existsUnique
/-- Shorthand for using projection notation with `Function.bijective_iff_existsUnique`. -/
protected theorem Bijective.existsUnique {f : α → β} (hf : Bijective f) (b : β) :
∃! a : α, f a = b :=
(bijective_iff_existsUnique f).mp hf b
#align function.bijective.exists_unique Function.Bijective.existsUnique
theorem Bijective.existsUnique_iff {f : α → β} (hf : Bijective f) {p : β → Prop} :
(∃! y, p y) ↔ ∃! x, p (f x) :=
⟨fun ⟨y, hpy, hy⟩ ↦
let ⟨x, hx⟩ := hf.surjective y
⟨x, by simpa [hx], fun z (hz : p (f z)) ↦ hf.injective <| hx.symm ▸ hy _ hz⟩,
fun ⟨x, hpx, hx⟩ ↦
⟨f x, hpx, fun y hy ↦
let ⟨z, hz⟩ := hf.surjective y
hz ▸ congr_arg f (hx _ (by simpa [hz]))⟩⟩
#align function.bijective.exists_unique_iff Function.Bijective.existsUnique_iff
theorem Bijective.of_comp_iff (f : α → β) {g : γ → α} (hg : Bijective g) :
Bijective (f ∘ g) ↔ Bijective f :=
and_congr (Injective.of_comp_iff' _ hg) (Surjective.of_comp_iff _ hg.surjective)
#align function.bijective.of_comp_iff Function.Bijective.of_comp_iff
theorem Bijective.of_comp_iff' {f : α → β} (hf : Bijective f) (g : γ → α) :
Function.Bijective (f ∘ g) ↔ Function.Bijective g :=
and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
#align function.bijective.of_comp_iff' Function.Bijective.of_comp_iff'
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
theorem cantor_surjective {α} (f : α → Set α) : ¬Surjective f
| h => let ⟨D, e⟩ := h {a | ¬ f a a}
@iff_not_self (D ∈ f D) <| iff_of_eq <| congr_arg (D ∈ ·) e
#align function.cantor_surjective Function.cantor_surjective
/-- **Cantor's diagonal argument** implies that there are no injective functions from `Set α`
to `α`. -/
theorem cantor_injective {α : Type*} (f : Set α → α) : ¬Injective f
| i => cantor_surjective (fun a ↦ {b | ∀ U, a = f U → U b}) <|
RightInverse.surjective (λ U => Set.ext <| fun _ ↦ ⟨fun h ↦ h U rfl, fun h _ e ↦ i e ▸ h⟩)
#align function.cantor_injective Function.cantor_injective
/-- There is no surjection from `α : Type u` into `Type (max u v)`. This theorem
demonstrates why `Type : Type` would be inconsistent in Lean. -/
theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f := by
intro hf
let T : Type max u v := Sigma f
cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
#align function.not_surjective_Type Function.not_surjective_Type
/-- `g` is a partial inverse to `f` (an injective but not necessarily
surjective function) if `g y = some x` implies `f x = y`, and `g y = none`
implies that `y` is not in the range of `f`. -/
def IsPartialInv {α β} (f : α → β) (g : β → Option α) : Prop :=
∀ x y, g y = some x ↔ f x = y
#align function.is_partial_inv Function.IsPartialInv
theorem isPartialInv_left {α β} {f : α → β} {g} (H : IsPartialInv f g) (x) : g (f x) = some x :=
(H _ _).2 rfl
#align function.is_partial_inv_left Function.isPartialInv_left
theorem injective_of_isPartialInv {α β} {f : α → β} {g} (H : IsPartialInv f g) :
Injective f := fun _ _ h ↦
Option.some.inj <| ((H _ _).2 h).symm.trans ((H _ _).2 rfl)
#align function.injective_of_partial_inv Function.injective_of_isPartialInv
theorem injective_of_isPartialInv_right {α β} {f : α → β} {g} (H : IsPartialInv f g) (x y b)
(h₁ : b ∈ g x) (h₂ : b ∈ g y) : x = y :=
((H _ _).1 h₁).symm.trans ((H _ _).1 h₂)
#align function.injective_of_partial_inv_right Function.injective_of_isPartialInv_right
theorem LeftInverse.comp_eq_id {f : α → β} {g : β → α} (h : LeftInverse f g) : f ∘ g = id :=
funext h
#align function.left_inverse.comp_eq_id Function.LeftInverse.comp_eq_id
theorem leftInverse_iff_comp {f : α → β} {g : β → α} : LeftInverse f g ↔ f ∘ g = id :=
⟨LeftInverse.comp_eq_id, congr_fun⟩
#align function.left_inverse_iff_comp Function.leftInverse_iff_comp
theorem RightInverse.comp_eq_id {f : α → β} {g : β → α} (h : RightInverse f g) : g ∘ f = id :=
funext h
#align function.right_inverse.comp_eq_id Function.RightInverse.comp_eq_id
theorem rightInverse_iff_comp {f : α → β} {g : β → α} : RightInverse f g ↔ g ∘ f = id :=
⟨RightInverse.comp_eq_id, congr_fun⟩
#align function.right_inverse_iff_comp Function.rightInverse_iff_comp
theorem LeftInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : LeftInverse f g)
(hh : LeftInverse h i) : LeftInverse (h ∘ f) (g ∘ i) :=
fun a ↦ show h (f (g (i a))) = a by rw [hf (i a), hh a]
#align function.left_inverse.comp Function.LeftInverse.comp
theorem RightInverse.comp {f : α → β} {g : β → α} {h : β → γ} {i : γ → β} (hf : RightInverse f g)
(hh : RightInverse h i) : RightInverse (h ∘ f) (g ∘ i) :=
LeftInverse.comp hh hf
#align function.right_inverse.comp Function.RightInverse.comp
theorem LeftInverse.rightInverse {f : α → β} {g : β → α} (h : LeftInverse g f) : RightInverse f g :=
h
#align function.left_inverse.right_inverse Function.LeftInverse.rightInverse
theorem RightInverse.leftInverse {f : α → β} {g : β → α} (h : RightInverse g f) : LeftInverse f g :=
h
#align function.right_inverse.left_inverse Function.RightInverse.leftInverse
theorem LeftInverse.surjective {f : α → β} {g : β → α} (h : LeftInverse f g) : Surjective f :=
h.rightInverse.surjective
#align function.left_inverse.surjective Function.LeftInverse.surjective
theorem RightInverse.injective {f : α → β} {g : β → α} (h : RightInverse f g) : Injective f :=
h.leftInverse.injective
#align function.right_inverse.injective Function.RightInverse.injective
theorem LeftInverse.rightInverse_of_injective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hf : Injective f) : RightInverse f g :=
fun x ↦ hf <| h (f x)
#align function.left_inverse.right_inverse_of_injective Function.LeftInverse.rightInverse_of_injective
theorem LeftInverse.rightInverse_of_surjective {f : α → β} {g : β → α} (h : LeftInverse f g)
(hg : Surjective g) : RightInverse f g :=
fun x ↦ let ⟨y, hy⟩ := hg x; hy ▸ congr_arg g (h y)
#align function.left_inverse.right_inverse_of_surjective Function.LeftInverse.rightInverse_of_surjective
theorem RightInverse.leftInverse_of_surjective {f : α → β} {g : β → α} :
RightInverse f g → Surjective f → LeftInverse f g :=
LeftInverse.rightInverse_of_surjective
#align function.right_inverse.left_inverse_of_surjective Function.RightInverse.leftInverse_of_surjective
theorem RightInverse.leftInverse_of_injective {f : α → β} {g : β → α} :
RightInverse f g → Injective g → LeftInverse f g :=
LeftInverse.rightInverse_of_injective
#align function.right_inverse.left_inverse_of_injective Function.RightInverse.leftInverse_of_injective
theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h₁ : LeftInverse g₁ f)
(h₂ : RightInverse g₂ f) : g₁ = g₂ :=
calc
g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp.right_id]
_ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, comp.left_id]
#align function.left_inverse.eq_right_inverse Function.LeftInverse.eq_rightInverse
attribute [local instance] Classical.propDecidable
/-- We can use choice to construct explicitly a partial inverse for
a given injective function `f`. -/
noncomputable def partialInv {α β} (f : α → β) (b : β) : Option α :=
if h : ∃ a, f a = b then some (Classical.choose h) else none
#align function.partial_inv Function.partialInv
theorem partialInv_of_injective {α β} {f : α → β} (I : Injective f) : IsPartialInv f (partialInv f)
| a, b =>
⟨fun h =>
have hpi : partialInv f b = if h : ∃ a, f a = b then some (Classical.choose h) else none :=
rfl
if h' : ∃ a, f a = b
then by rw [hpi, dif_pos h'] at h
injection h with h
subst h
apply Classical.choose_spec h'
else by rw [hpi, dif_neg h'] at h; contradiction,
fun e => e ▸ have h : ∃ a', f a' = f a := ⟨_, rfl⟩
(dif_pos h).trans (congr_arg _ (I <| Classical.choose_spec h))⟩
#align function.partial_inv_of_injective Function.partialInv_of_injective
theorem partialInv_left {α β} {f : α → β} (I : Injective f) : ∀ x, partialInv f (f x) = some x :=
isPartialInv_left (partialInv_of_injective I)
#align function.partial_inv_left Function.partialInv_left
end
section InvFun
variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
attribute [local instance] Classical.propDecidable
/-- The inverse of a function (which is a left inverse if `f` is injective
and a right inverse if `f` is surjective). -/
-- Explicit Sort so that `α` isn't inferred to be Prop via `exists_prop_decidable`
noncomputable def invFun {α : Sort u} {β} [Nonempty α] (f : α → β) : β → α :=
fun y ↦ if h : (∃ x, f x = y) then h.choose else Classical.arbitrary α
#align function.inv_fun Function.invFun
theorem invFun_eq (h : ∃ a, f a = b) : f (invFun f b) = b :=
by simp only [invFun, dif_pos h, h.choose_spec]
#align function.inv_fun_eq Function.invFun_eq
theorem apply_invFun_apply {α : Type u₁} {β : Type u₂} {f : α → β} {a : α} :
f (@invFun _ _ ⟨a⟩ f (f a)) = f a :=
@invFun_eq _ _ ⟨a⟩ _ _ ⟨_, rfl⟩
theorem invFun_neg (h : ¬∃ a, f a = b) : invFun f b = Classical.choice ‹_› :=
dif_neg h
#align function.inv_fun_neg Function.invFun_neg
theorem invFun_eq_of_injective_of_rightInverse {g : β → α} (hf : Injective f)
(hg : RightInverse g f) : invFun f = g :=
funext fun b ↦
hf
(by
rw [hg b]
exact invFun_eq ⟨g b, hg b⟩)
#align function.inv_fun_eq_of_injective_of_right_inverse Function.invFun_eq_of_injective_of_rightInverse
theorem rightInverse_invFun (hf : Surjective f) : RightInverse (invFun f) f :=
fun b ↦ invFun_eq <| hf b
#align function.right_inverse_inv_fun Function.rightInverse_invFun
theorem leftInverse_invFun (hf : Injective f) : LeftInverse (invFun f) f :=
fun b ↦ hf <| invFun_eq ⟨b, rfl⟩
#align function.left_inverse_inv_fun Function.leftInverse_invFun
theorem invFun_surjective (hf : Injective f) : Surjective (invFun f) :=
(leftInverse_invFun hf).surjective
#align function.inv_fun_surjective Function.invFun_surjective
theorem invFun_comp (hf : Injective f) : invFun f ∘ f = id :=
funext <| leftInverse_invFun hf
#align function.inv_fun_comp Function.invFun_comp
theorem Injective.hasLeftInverse (hf : Injective f) : HasLeftInverse f :=
⟨invFun f, leftInverse_invFun hf⟩
#align function.injective.has_left_inverse Function.Injective.hasLeftInverse
theorem injective_iff_hasLeftInverse : Injective f ↔ HasLeftInverse f :=
⟨Injective.hasLeftInverse, HasLeftInverse.injective⟩
#align function.injective_iff_has_left_inverse Function.injective_iff_hasLeftInverse
end InvFun
section SurjInv
variable {α : Sort u} {β : Sort v} {γ : Sort w} {f : α → β}
/-- The inverse of a surjective function. (Unlike `invFun`, this does not require
`α` to be inhabited.) -/
noncomputable def surjInv {f : α → β} (h : Surjective f) (b : β) : α :=
Classical.choose (h b)
#align function.surj_inv Function.surjInv
theorem surjInv_eq (h : Surjective f) (b) : f (surjInv h b) = b :=
Classical.choose_spec (h b)
#align function.surj_inv_eq Function.surjInv_eq
theorem rightInverse_surjInv (hf : Surjective f) : RightInverse (surjInv hf) f :=
surjInv_eq hf
#align function.right_inverse_surj_inv Function.rightInverse_surjInv
theorem leftInverse_surjInv (hf : Bijective f) : LeftInverse (surjInv hf.2) f :=
rightInverse_of_injective_of_leftInverse hf.1 (rightInverse_surjInv hf.2)
#align function.left_inverse_surj_inv Function.leftInverse_surjInv
theorem Surjective.hasRightInverse (hf : Surjective f) : HasRightInverse f :=
⟨_, rightInverse_surjInv hf⟩
#align function.surjective.has_right_inverse Function.Surjective.hasRightInverse
theorem surjective_iff_hasRightInverse : Surjective f ↔ HasRightInverse f :=
⟨Surjective.hasRightInverse, HasRightInverse.surjective⟩
#align function.surjective_iff_has_right_inverse Function.surjective_iff_hasRightInverse
theorem bijective_iff_has_inverse : Bijective f ↔ ∃ g, LeftInverse g f ∧ RightInverse g f :=
⟨fun hf ↦ ⟨_, leftInverse_surjInv hf, rightInverse_surjInv hf.2⟩, fun ⟨_, gl, gr⟩ ↦
⟨gl.injective, gr.surjective⟩⟩
#align function.bijective_iff_has_inverse Function.bijective_iff_has_inverse
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
(rightInverse_surjInv h).injective
#align function.injective_surj_inv Function.injective_surjInv
theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α → β) :
Surjective f :=
fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
#align function.surjective_to_subsingleton Function.surjective_to_subsingleton
/-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
#align function.surjective.comp_left Function.Surjective.comp_left
/-- Composition by a bijective function on the left is itself bijective. -/
theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
Bijective (g ∘ · : (α → β) → α → γ) :=
⟨hg.injective.comp_left, hg.surjective.comp_left⟩
#align function.bijective.comp_left Function.Bijective.comp_left
end SurjInv
section Update
variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α] [DecidableEq α']
{f g : (a : α) → β a} {a : α} {b : β a}
/-- Replacing the value of a function at a given point by a given value. -/
def update (f : ∀ a, β a) (a' : α) (v : β a') (a : α) : β a :=
if h : a = a' then Eq.ndrec v h.symm else f a
#align function.update Function.update
@[simp]
theorem update_same (a : α) (v : β a) (f : ∀ a, β a) : update f a v a = v :=
dif_pos rfl
#align function.update_same Function.update_same
@[simp]
theorem update_noteq {a a' : α} (h : a ≠ a') (v : β a') (f : ∀ a, β a) : update f a' v a = f a :=
dif_neg h
#align function.update_noteq Function.update_noteq
/-- On non-dependent functions, `Function.update` can be expressed as an `ite` -/
theorem update_apply {β : Sort*} (f : α → β) (a' : α) (b : β) (a : α) :
update f a' b a = if a = a' then b else f a := by
rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*]
#align function.update_apply Function.update_apply
@[nontriviality]
theorem update_eq_const_of_subsingleton [Subsingleton α] (a : α) (v : α') (f : α → α') :
update f a v = const α v :=
funext fun a' ↦ Subsingleton.elim a a' ▸ update_same _ _ _
theorem surjective_eval {α : Sort u} {β : α → Sort v} [h : ∀ a, Nonempty (β a)] (a : α) :
Surjective (eval a : (∀ a, β a) → β a) := fun b ↦
⟨@update _ _ (Classical.decEq α) (fun a ↦ (h a).some) a b,
@update_same _ _ (Classical.decEq α) _ _ _⟩
#align function.surjective_eval Function.surjective_eval
theorem update_injective (f : ∀ a, β a) (a' : α) : Injective (update f a') := fun v v' h ↦ by
have := congr_fun h a'
rwa [update_same, update_same] at this
#align function.update_injective Function.update_injective
lemma forall_update_iff (f : ∀a, β a) {a : α} {b : β a} (p : ∀a, β a → Prop) :
(∀ x, p x (update f a b x)) ↔ p a b ∧ ∀ x, x ≠ a → p x (f x) := by
rw [← and_forall_ne a, update_same]
simp (config := { contextual := true })
#align function.forall_update_iff Function.forall_update_iff
theorem exists_update_iff (f : ∀ a, β a) {a : α} {b : β a} (p : ∀ a, β a → Prop) :
(∃ x, p x (update f a b x)) ↔ p a b ∨ ∃ (x : _) (_ : x ≠ a), p x (f x) := by
rw [← not_forall_not, forall_update_iff f fun a b ↦ ¬p a b]
simp [-not_and, not_and_or]
#align function.exists_update_iff Function.exists_update_iff
theorem update_eq_iff {a : α} {b : β a} {f g : ∀ a, β a} :
update f a b = g ↔ b = g a ∧ ∀ (x) (_ : x ≠ a), f x = g x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ y = g x
#align function.update_eq_iff Function.update_eq_iff
theorem eq_update_iff {a : α} {b : β a} {f g : ∀ a, β a} :
g = update f a b ↔ g a = b ∧ ∀ (x) (_ : x ≠ a), g x = f x :=
funext_iff.trans <| forall_update_iff _ fun x y ↦ g x = y
#align function.eq_update_iff Function.eq_update_iff
@[simp] lemma update_eq_self_iff : update f a b = f ↔ b = f a := by simp [update_eq_iff]
#align function.update_eq_self_iff Function.update_eq_self_iff
@[simp] lemma eq_update_self_iff : f = update f a b ↔ f a = b := by simp [eq_update_iff]
#align function.eq_update_self_iff Function.eq_update_self_iff
lemma ne_update_self_iff : f ≠ update f a b ↔ f a ≠ b := eq_update_self_iff.not
#align function.ne_update_self_iff Function.ne_update_self_iff
lemma update_ne_self_iff : update f a b ≠ f ↔ b ≠ f a := update_eq_self_iff.not
#align function.update_ne_self_iff Function.update_ne_self_iff
@[simp]
theorem update_eq_self (a : α) (f : ∀ a, β a) : update f a (f a) = f :=
update_eq_iff.2 ⟨rfl, fun _ _ ↦ rfl⟩
#align function.update_eq_self Function.update_eq_self
theorem update_comp_eq_of_forall_ne' {α'} (g : ∀ a, β a) {f : α' → α} {i : α} (a : β i)
(h : ∀ x, f x ≠ i) : (fun j ↦ (update g i a) (f j)) = fun j ↦ g (f j) :=
funext fun _ ↦ update_noteq (h _) _ _
#align function.update_comp_eq_of_forall_ne' Function.update_comp_eq_of_forall_ne'
/-- Non-dependent version of `Function.update_comp_eq_of_forall_ne'` -/
theorem update_comp_eq_of_forall_ne {α β : Sort*} (g : α' → β) {f : α → α'} {i : α'} (a : β)
(h : ∀ x, f x ≠ i) : update g i a ∘ f = g ∘ f :=
update_comp_eq_of_forall_ne' g a h
#align function.update_comp_eq_of_forall_ne Function.update_comp_eq_of_forall_ne
theorem update_comp_eq_of_injective' (g : ∀ a, β a) {f : α' → α} (hf : Function.Injective f)
(i : α') (a : β (f i)) : (fun j ↦ update g (f i) a (f j)) = update (fun i ↦ g (f i)) i a :=
eq_update_iff.2 ⟨update_same _ _ _, fun _ hj ↦ update_noteq (hf.ne hj) _ _⟩
#align function.update_comp_eq_of_injective' Function.update_comp_eq_of_injective'
/-- Non-dependent version of `Function.update_comp_eq_of_injective'` -/
theorem update_comp_eq_of_injective {β : Sort*} (g : α' → β) {f : α → α'}
(hf : Function.Injective f) (i : α) (a : β) :
Function.update g (f i) a ∘ f = Function.update (g ∘ f) i a :=
update_comp_eq_of_injective' g hf i a
#align function.update_comp_eq_of_injective Function.update_comp_eq_of_injective
theorem apply_update {ι : Sort*} [DecidableEq ι] {α β : ι → Sort*} (f : ∀ i, α i → β i)
(g : ∀ i, α i) (i : ι) (v : α i) (j : ι) :
f j (update g i v j) = update (fun k ↦ f k (g k)) i (f i v) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update Function.apply_update
theorem apply_update₂ {ι : Sort*} [DecidableEq ι] {α β γ : ι → Sort*} (f : ∀ i, α i → β i → γ i)
(g : ∀ i, α i) (h : ∀ i, β i) (i : ι) (v : α i) (w : β i) (j : ι) :
f j (update g i v j) (update h i w j) = update (fun k ↦ f k (g k) (h k)) i (f i v w) j := by
by_cases h:j = i
· subst j
simp
· simp [h]
#align function.apply_update₂ Function.apply_update₂
theorem pred_update (P : ∀ ⦃a⦄, β a → Prop) (f : ∀ a, β a) (a' : α) (v : β a') (a : α) :
P (update f a' v a) ↔ a = a' ∧ P v ∨ a ≠ a' ∧ P (f a) := by
rw [apply_update P, update_apply, ite_prop_iff_or]
theorem comp_update {α' : Sort*} {β : Sort*} (f : α' → β) (g : α → α') (i : α) (v : α') :
f ∘ update g i v = update (f ∘ g) i (f v) :=
funext <| apply_update _ _ _ _
#align function.comp_update Function.comp_update
theorem update_comm {α} [DecidableEq α] {β : α → Sort*} {a b : α} (h : a ≠ b) (v : β a) (w : β b)
(f : ∀ a, β a) : update (update f a v) b w = update (update f b w) a v := by
funext c
simp only [update]
by_cases h₁ : c = b <;> by_cases h₂ : c = a
· rw [dif_pos h₁, dif_pos h₂]
cases h (h₂.symm.trans h₁)
· rw [dif_pos h₁, dif_pos h₁, dif_neg h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_pos h₂]
· rw [dif_neg h₁, dif_neg h₁, dif_neg h₂]
#align function.update_comm Function.update_comm
@[simp]
theorem update_idem {α} [DecidableEq α] {β : α → Sort*} {a : α} (v w : β a) (f : ∀ a, β a) :
update (update f a v) a w = update f a w := by
funext b
by_cases h : b = a <;> simp [update, h]
#align function.update_idem Function.update_idem
end Update
noncomputable section Extend
attribute [local instance] Classical.propDecidable
variable {α β γ : Sort*} {f : α → β}
/-- Extension of a function `g : α → γ` along a function `f : α → β`.
For every `a : α`, `f a` is sent to `g a`. `f` might not be surjective, so we use an auxiliary
function `j : β → γ` by sending `b : β` not in the range of `f` to `j b`. If you do not care about
the behavior outside the range, `j` can be used as a junk value by setting it to be `0` or
`Classical.arbitrary` (assuming `γ` is nonempty).
This definition is mathematically meaningful only when `f a₁ = f a₂ → g a₁ = g a₂` (spelled
`g.FactorsThrough f`). In particular this holds if `f` is injective.
A typical use case is extending a function from a subtype to the entire type. If you wish to extend
`g : {b : β // p b} → γ` to a function `β → γ`, you should use `Function.extend Subtype.val g j`. -/
def extend (f : α → β) (g : α → γ) (j : β → γ) : β → γ := fun b ↦
if h : ∃ a, f a = b then g (Classical.choose h) else j b
#align function.extend Function.extend
/-- g factors through f : `f a = f b → g a = g b` -/
def FactorsThrough (g : α → γ) (f : α → β) : Prop :=
∀ ⦃a b⦄, f a = f b → g a = g b
#align function.factors_through Function.FactorsThrough
theorem extend_def (f : α → β) (g : α → γ) (e' : β → γ) (b : β) [Decidable (∃ a, f a = b)] :
extend f g e' b = if h : ∃ a, f a = b then g (Classical.choose h) else e' b := by
unfold extend
congr
#align function.extend_def Function.extend_def
lemma Injective.factorsThrough (hf : Injective f) (g : α → γ) : g.FactorsThrough f :=
fun _ _ h => congr_arg g (hf h)
#align function.injective.factors_through Function.Injective.factorsThrough
lemma FactorsThrough.extend_apply {g : α → γ} (hf : g.FactorsThrough f) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a := by
simp only [extend_def, dif_pos, exists_apply_eq_apply]
exact hf (Classical.choose_spec (exists_apply_eq_apply f a))
#align function.factors_through.extend_apply Function.FactorsThrough.extend_apply
@[simp]
theorem Injective.extend_apply (hf : Injective f) (g : α → γ) (e' : β → γ) (a : α) :
extend f g e' (f a) = g a :=
(hf.factorsThrough g).extend_apply e' a
#align function.injective.extend_apply Function.Injective.extend_apply
@[simp]
theorem extend_apply' (g : α → γ) (e' : β → γ) (b : β) (hb : ¬∃ a, f a = b) :
extend f g e' b = e' b := by
simp [Function.extend_def, hb]
#align function.extend_apply' Function.extend_apply'
lemma factorsThrough_iff (g : α → γ) [Nonempty γ] : g.FactorsThrough f ↔ ∃ (e : β → γ), g = e ∘ f :=
⟨fun hf => ⟨extend f g (const β (Classical.arbitrary γ)),
funext (fun x => by simp only [comp_apply, hf.extend_apply])⟩,
fun h _ _ hf => by rw [Classical.choose_spec h, comp_apply, comp_apply, hf]⟩
#align function.factors_through_iff Function.factorsThrough_iff
lemma apply_extend {δ} {g : α → γ} (F : γ → δ) (f : α → β) (e' : β → γ) (b : β) :
F (extend f g e' b) = extend f (F ∘ g) (F ∘ e') b :=
apply_dite F _ _ _
#align function.factors_through.apply_extend Function.apply_extend
#align function.injective.apply_extend Function.apply_extend
theorem extend_injective (hf : Injective f) (e' : β → γ) : Injective fun g ↦ extend f g e' := by
intro g₁ g₂ hg
refine' funext fun x ↦ _
have H := congr_fun hg (f x)
simp only [hf.extend_apply] at H
exact H
#align function.extend_injective Function.extend_injective
lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsThrough g f) :
extend f g e' ∘ f = g :=
funext $ fun a => hf.extend_apply e' a
#align function.factors_through.extend_comp Function.FactorsThrough.extend_comp
@[simp]
theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
funext fun a ↦ hf.extend_apply g e' a
#align function.extend_comp Function.extend_comp
theorem Injective.surjective_comp_right' (hf : Injective f) (g₀ : β → γ) :
Surjective fun g : β → γ ↦ g ∘ f :=
fun g ↦ ⟨extend f g g₀, extend_comp hf _ _⟩
#align function.injective.surjective_comp_right' Function.Injective.surjective_comp_right'
theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
Surjective fun g : β → γ ↦ g ∘ f :=
hf.surjective_comp_right' fun _ ↦ Classical.choice ‹_›
#align function.injective.surjective_comp_right Function.Injective.surjective_comp_right
theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
⟨hf.surjective.injective_comp_right, fun g ↦
⟨g ∘ surjInv hf.surjective,
by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp.right_id]⟩⟩
#align function.bijective.comp_right Function.Bijective.comp_right
end Extend
theorem uncurry_def {α β γ} (f : α → β → γ) : uncurry f = fun p ↦ f p.1 p.2 :=
rfl
#align function.uncurry_def Function.uncurry_def
@[simp]
theorem uncurry_apply_pair {α β γ} (f : α → β → γ) (x : α) (y : β) : uncurry f (x, y) = f x y :=
rfl
#align function.uncurry_apply_pair Function.uncurry_apply_pair
@[simp]
theorem curry_apply {α β γ} (f : α × β → γ) (x : α) (y : β) : curry f x y = f (x, y) :=
rfl
#align function.curry_apply Function.curry_apply
section Bicomp
variable {α β γ δ ε : Type*}
/-- Compose a binary function `f` with a pair of unary functions `g` and `h`.
If both arguments of `f` have the same type and `g = h`, then `bicompl f g g = f on g`. -/
def bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) (a b) :=
f (g a) (h b)
#align function.bicompl Function.bicompl
/-- Compose a unary function `f` with a binary function `g`. -/
def bicompr (f : γ → δ) (g : α → β → γ) (a b) :=
f (g a b)
#align function.bicompr Function.bicompr
-- Suggested local notation:
local notation f " ∘₂ " g => bicompr f g
theorem uncurry_bicompr (f : α → β → γ) (g : γ → δ) : uncurry (g ∘₂ f) = g ∘ uncurry f :=
rfl
#align function.uncurry_bicompr Function.uncurry_bicompr
theorem uncurry_bicompl (f : γ → δ → ε) (g : α → γ) (h : β → δ) :
uncurry (bicompl f g h) = uncurry f ∘ Prod.map g h :=
rfl
#align function.uncurry_bicompl Function.uncurry_bicompl
end Bicomp
section Uncurry
variable {α β γ δ : Type*}
/-- Records a way to turn an element of `α` into a function from `β` to `γ`. The most generic use
is to recursively uncurry. For instance `f : α → β → γ → δ` will be turned into
`↿f : α × β × γ → δ`. One can also add instances for bundled maps. -/
class HasUncurry (α : Type*) (β : outParam (Type*)) (γ : outParam (Type*)) where
/-- Uncurrying operator. The most generic use is to recursively uncurry. For instance
`f : α → β → γ → δ` will be turned into `↿f : α × β × γ → δ`. One can also add instances
for bundled maps.-/
uncurry : α → β → γ
#align function.has_uncurry Function.HasUncurry
notation:arg "↿" x:arg => HasUncurry.uncurry x
instance hasUncurryBase : HasUncurry (α → β) α β :=
⟨id⟩
instance hasUncurryInduction [HasUncurry β γ δ] : HasUncurry (α → β) (α × γ) δ :=
⟨fun f p ↦ (↿(f p.1)) p.2⟩
end Uncurry
/-- A function is involutive, if `f ∘ f = id`. -/
def Involutive {α} (f : α → α) : Prop :=
∀ x, f (f x) = x
#align function.involutive Function.Involutive
theorem _root_.Bool.involutive_not : Involutive not :=
Bool.not_not
namespace Involutive
variable {α : Sort u} {f : α → α} (h : Involutive f)
@[simp]
theorem comp_self : f ∘ f = id :=
funext h
#align function.involutive.comp_self Function.Involutive.comp_self
protected theorem leftInverse : LeftInverse f f := h
#align function.involutive.left_inverse Function.Involutive.leftInverse
protected theorem rightInverse : RightInverse f f := h
#align function.involutive.right_inverse Function.Involutive.rightInverse
protected theorem injective : Injective f := h.leftInverse.injective
#align function.involutive.injective Function.Involutive.injective
protected theorem surjective : Surjective f := fun x ↦ ⟨f x, h x⟩
#align function.involutive.surjective Function.Involutive.surjective
protected theorem bijective : Bijective f := ⟨h.injective, h.surjective⟩
#align function.involutive.bijective Function.Involutive.bijective
/-- Involuting an `ite` of an involuted value `x : α` negates the `Prop` condition in the `ite`. -/
protected theorem ite_not (P : Prop) [Decidable P] (x : α) : f (ite P x (f x)) = ite (¬P) x (f x) :=
by rw [apply_ite f, h, ite_not]
#align function.involutive.ite_not Function.Involutive.ite_not
/-- An involution commutes across an equality. Compare to `Function.Injective.eq_iff`. -/
protected theorem eq_iff {x y : α} : f x = y ↔ x = f y :=
h.injective.eq_iff' (h y)
#align function.involutive.eq_iff Function.Involutive.eq_iff
end Involutive
@[simp]
lemma symmetric_apply_eq_iff {f : α → α} : Symmetric (f · = ·) ↔ Involutive f := by
simp [Symmetric, Involutive]
/-- The property of a binary function `f : α → β → γ` being injective.
Mathematically this should be thought of as the corresponding function `α × β → γ` being injective.
-/
def Injective2 {α β γ} (f : α → β → γ) : Prop :=
∀ ⦃a₁ a₂ b₁ b₂⦄, f a₁ b₁ = f a₂ b₂ → a₁ = a₂ ∧ b₁ = b₂
#align function.injective2 Function.Injective2
namespace Injective2
variable {α β γ : Sort*} {f : α → β → γ}
/-- A binary injective function is injective when only the left argument varies. -/
protected theorem left (hf : Injective2 f) (b : β) : Function.Injective fun a ↦ f a b :=
fun _ _ h ↦ (hf h).left
#align function.injective2.left Function.Injective2.left
/-- A binary injective function is injective when only the right argument varies. -/
protected theorem right (hf : Injective2 f) (a : α) : Function.Injective (f a) :=
fun _ _ h ↦ (hf h).right
#align function.injective2.right Function.Injective2.right
protected theorem uncurry {α β γ : Type*} {f : α → β → γ} (hf : Injective2 f) :
Function.Injective (uncurry f) :=
fun ⟨_, _⟩ ⟨_, _⟩ h ↦ (hf h).elim (congr_arg₂ _)
#align function.injective2.uncurry Function.Injective2.uncurry
/-- As a map from the left argument to a unary function, `f` is injective. -/
theorem left' (hf : Injective2 f) [Nonempty β] : Function.Injective f := fun a₁ a₂ h ↦
let ⟨b⟩ := ‹Nonempty β›
hf.left b <| (congr_fun h b : _)
#align function.injective2.left' Function.Injective2.left'
/-- As a map from the right argument to a unary function, `f` is injective. -/
theorem right' (hf : Injective2 f) [Nonempty α] : Function.Injective fun b a ↦ f a b :=
fun b₁ b₂ h ↦
let ⟨a⟩ := ‹Nonempty α›
hf.right a <| (congr_fun h a : _)
#align function.injective2.right' Function.Injective2.right'
theorem eq_iff (hf : Injective2 f) {a₁ a₂ b₁ b₂} : f a₁ b₁ = f a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ :=
⟨fun h ↦ hf h, fun ⟨h1, h2⟩ ↦ congr_arg₂ f h1 h2⟩
#align function.injective2.eq_iff Function.Injective2.eq_iff
end Injective2
section Sometimes
attribute [local instance] Classical.propDecidable
/-- `sometimes f` evaluates to some value of `f`, if it exists. This function is especially
interesting in the case where `α` is a proposition, in which case `f` is necessarily a
constant function, so that `sometimes f = f a` for all `a`. -/
noncomputable def sometimes {α β} [Nonempty β] (f : α → β) : β :=
if h : Nonempty α then f (Classical.choice h) else Classical.choice ‹_›
#align function.sometimes Function.sometimes
theorem sometimes_eq {p : Prop} {α} [Nonempty α] (f : p → α) (a : p) : sometimes f = f a :=
dif_pos ⟨a⟩
#align function.sometimes_eq Function.sometimes_eq
theorem sometimes_spec {p : Prop} {α} [Nonempty α] (P : α → Prop) (f : p → α) (a : p)
(h : P (f a)) : P (sometimes f) :=
by rwa [sometimes_eq]
#align function.sometimes_spec Function.sometimes_spec
end Sometimes
end Function
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
refine ⟨fun h ↦ ?_, ?_⟩
· refine ⟨fun a ↦ (h a).choose, fun hr ↦ ?_, fun h' ↦ h' ▸ ?_⟩
exacts [((h _).choose_spec.2 _ hr).symm, (h _).choose_spec.1]
· rintro ⟨f, hf⟩
simp [hf]
/-- A relation `r : α → β → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff' {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, r = (f · = ·) := by
simp [forall_existsUnique_iff, Function.funext_iff]
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff' {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ r = (f · = ·) := by
refine ⟨fun h ↦ ?_, fun ⟨f, _, hf⟩ ↦ forall_existsUnique_iff'.2 ⟨f, hf⟩⟩
rcases forall_existsUnique_iff'.1 h with ⟨f, rfl : r = _⟩
exact ⟨f, symmetric_apply_eq_iff.1 hr, rfl⟩
/-- A symmetric relation `r : α → α → Prop` is "function-like"
(for each `a` there exists a unique `b` such that `r a b`)
if and only if it is `(f · = ·)` for some involutive function `f`. -/
protected lemma Symmetric.forall_existsUnique_iff {r : α → α → Prop} (hr : Symmetric r) :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → α, Involutive f ∧ ∀ {a b}, r a b ↔ f a = b := by
simp [hr.forall_existsUnique_iff', funext_iff]
/-- `s.piecewise f g` is the function equal to `f` on the set `s`, and to `g` on its complement. -/
def Set.piecewise {α : Type u} {β : α → Sort v} (s : Set α) (f g : ∀ i, β i)
[∀ j, Decidable (j ∈ s)] : ∀ i, β i :=
fun i ↦ if i ∈ s then f i else g i
#align set.piecewise Set.piecewise
/-! ### Bijectivity of `Eq.rec`, `Eq.mp`, `Eq.mpr`, and `cast` -/
theorem eq_rec_on_bijective {α : Sort*} {C : α → Sort*} :
∀ {a a' : α} (h : a = a'), Function.Bijective (@Eq.ndrec _ _ C · _ h)
| _, _, rfl => ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_rec_on_bijective eq_rec_on_bijective
theorem eq_mp_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mp h) := by
-- TODO: mathlib3 uses `eq_rec_on_bijective`, difference in elaboration here
-- due to `@[macro_inline]` possibly?
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mp_bijective eq_mp_bijective
theorem eq_mpr_bijective {α β : Sort _} (h : α = β) : Function.Bijective (Eq.mpr h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align eq_mpr_bijective eq_mpr_bijective
theorem cast_bijective {α β : Sort _} (h : α = β) : Function.Bijective (cast h) := by
cases h
refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩
#align cast_bijective cast_bijective
/-! Note these lemmas apply to `Type*` not `Sort*`, as the latter interferes with `simp`, and
is trivial anyway.-/
@[simp]
theorem eq_rec_inj {α : Sort*} {a a' : α} (h : a = a') {C : α → Type*} (x y : C a) :
(Eq.ndrec x h : C a') = Eq.ndrec y h ↔ x = y :=
(eq_rec_on_bijective h).injective.eq_iff
#align eq_rec_inj eq_rec_inj
@[simp]
theorem cast_inj {α β : Type u} (h : α = β) {x y : α} : cast h x = cast h y ↔ x = y :=
(cast_bijective h).injective.eq_iff
#align cast_inj cast_inj
theorem Function.LeftInverse.eq_rec_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
@Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a :=
eq_of_heq <| (eq_rec_heq _ _).trans <| by rw [h]
#align function.left_inverse.eq_rec_eq Function.LeftInverse.eq_rec_eq
theorem Function.LeftInverse.eq_rec_on_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
-- TODO: mathlib3 uses `(congr_arg f (h a)).recOn (C (g (f a)))` for LHS
@Eq.recOn β (f (g (f a))) (fun x _ ↦ γ x) (f a) (congr_arg f (h a)) (C (g (f a))) = C a :=
h.eq_rec_eq _ _
#align function.left_inverse.eq_rec_on_eq Function.LeftInverse.eq_rec_on_eq
theorem Function.LeftInverse.cast_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
cast (congr_arg (fun a ↦ γ (f a)) (h a)) (C (g (f a))) = C a := by
| rw [cast_eq_iff_heq, h] | theorem Function.LeftInverse.cast_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
cast (congr_arg (fun a ↦ γ (f a)) (h a)) (C (g (f a))) = C a := by
| Mathlib.Logic.Function.Basic.1088_0.QX1TCPxnrBJfF8i | theorem Function.LeftInverse.cast_eq {α β : Sort*} {γ : β → Sort v} {f : α → β} {g : β → α}
(h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
cast (congr_arg (fun a ↦ γ (f a)) (h a)) (C (g (f a))) = C a | Mathlib_Logic_Function_Basic |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
J : Type v
inst✝ : Category.{v, v} J
F : J ⥤ SheafedSpace C
X Y : SheafedSpace C
f g : X ⟶ Y
⊢ Epi (coequalizer.π f g).base | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
| erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)] | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.48_0.tE6q65npbp8AX2g | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
J : Type v
inst✝ : Category.{v, v} J
F : J ⥤ SheafedSpace C
X Y : SheafedSpace C
f g : X ⟶ Y
⊢ Epi (coequalizer.π ((forget C).map f) ((forget C).map g) ≫ coequalizerComparison f g (forget C)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
| rw [← PreservesCoequalizer.iso_hom] | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.48_0.tE6q65npbp8AX2g | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasLimits C
J : Type v
inst✝ : Category.{v, v} J
F : J ⥤ SheafedSpace C
X Y : SheafedSpace C
f g : X ⟶ Y
⊢ Epi (coequalizer.π ((forget C).map f) ((forget C).map g) ≫ (PreservesCoequalizer.iso (forget C) f g).hom) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
| apply epi_comp | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.48_0.tE6q65npbp8AX2g | instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
x : ↑↑(colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace
⊢ LocalRing ↑(TopCat.Presheaf.stalk (colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace.presheaf x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
| obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.74_0.tE6q65npbp8AX2g | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
⊢ LocalRing
↑(TopCat.Presheaf.stalk (colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace.presheaf
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
| haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _ | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.74_0.tE6q65npbp8AX2g | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
this : LocalRing ↑(PresheafedSpace.stalk ((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace y)
⊢ LocalRing
↑(TopCat.Presheaf.stalk (colimit (F ⋙ forgetToSheafedSpace)).toPresheafedSpace.presheaf
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
| exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.74_0.tE6q65npbp8AX2g | /-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
x✝² x✝¹ : Discrete ι
j j' : ι
x✝ : { as := j } ⟶ { as := j' }
f : j = j'
⊢ F.map { down := { down := f } } ≫
(fun j =>
{ val := colimit.ι (F ⋙ forgetToSheafedSpace) j,
prop :=
(_ :
∀ (x : ↑↑(F.obj j).toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) j) x)) })
{ as := j' } =
(fun j =>
{ val := colimit.ι (F ⋙ forgetToSheafedSpace) j,
prop :=
(_ :
∀ (x : ↑↑(F.obj j).toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) j) x)) })
{ as := j } ≫
((Functor.const (Discrete ι)).obj (coproduct F)).map { down := { down := f } } | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by | subst f | /-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.88_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
x✝² x✝¹ : Discrete ι
j : ι
x✝ : { as := j } ⟶ { as := j }
⊢ F.map { down := { down := (_ : j = j) } } ≫
(fun j =>
{ val := colimit.ι (F ⋙ forgetToSheafedSpace) j,
prop :=
(_ :
∀ (x : ↑↑(F.obj j).toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) j) x)) })
{ as := j } =
(fun j =>
{ val := colimit.ι (F ⋙ forgetToSheafedSpace) j,
prop :=
(_ :
∀ (x : ↑↑(F.obj j).toPresheafedSpace),
IsLocalRingHom (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) j) x)) })
{ as := j } ≫
((Functor.const (Discrete ι)).obj (coproduct F)).map { down := { down := (_ : j = j) } } | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; | aesop | /-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; | Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.88_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
⊢ ∀ (x : ↑↑(coproductCofan F).pt.toPresheafedSpace),
IsLocalRingHom
(PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
| intro x | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
x : ↑↑(coproductCofan F).pt.toPresheafedSpace
⊢ IsLocalRingHom
(PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) x) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
| obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
⊢ IsLocalRingHom
(PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
| have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
this :
PresheafedSpace.stalkMap
(colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
y =
PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y) ≫
PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y
⊢ IsLocalRingHom
(PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
| rw [← IsIso.comp_inv_eq] at this | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
this :
PresheafedSpace.stalkMap
(colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
y ≫
inv (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y) =
PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)
⊢ IsLocalRingHom
(PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
| erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)] | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
this :
PresheafedSpace.stalkMap
(colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
y ≫
inv (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y) =
PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)
⊢ IsLocalRingHom
((eqToHom
(_ :
PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace
((colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).base
y) =
PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace
(((forgetToSheafedSpace.mapCocone s).ι.app i).base y)) ≫
PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) ≫
inv (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
| haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
case intro.intro.refl
ι : Type u
F : Discrete ι ⥤ LocallyRingedSpace
s : Cocone F
i : Discrete ι
y : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace
this✝ :
PresheafedSpace.stalkMap
(colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
y ≫
inv (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y) =
PresheafedSpace.stalkMap (colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s))
((colimit.ι (F ⋙ forgetToSheafedSpace) i).base y)
this : IsLocalRingHom (PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y)
⊢ IsLocalRingHom
((eqToHom
(_ :
PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace
((colimit.ι (F ⋙ forgetToSheafedSpace) i ≫
colimit.desc (F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)).base
y) =
PresheafedSpace.stalk (forgetToSheafedSpace.mapCocone s).pt.toPresheafedSpace
(((forgetToSheafedSpace.mapCocone s).ι.app i).base y)) ≫
PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) ≫
inv (PresheafedSpace.stalkMap (colimit.ι (F ⋙ forgetToSheafedSpace) i) y)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
| infer_instance | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.97_0.tE6q65npbp8AX2g | /-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
⊢ IsLocalRingHom ((coequalizer.π f.val g.val).c.app (op U)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
| have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
coequalizerComparison f.val g.val SheafedSpace.forgetToPresheafedSpace =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)
⊢ IsLocalRingHom ((coequalizer.π f.val g.val).c.app (op U)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
| rw [← PreservesCoequalizer.iso_hom] at this | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)
⊢ IsLocalRingHom ((coequalizer.π f.val g.val).c.app (op U)) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
| erw [SheafedSpace.congr_app this.symm (op U)] | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace
this :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)
⊢ IsLocalRingHom
((coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val) (SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).c.app
(op U) ≫
Y.presheaf.map
(eqToHom
(_ :
(Opens.map
(coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map f.val)
(SheafedSpace.forgetToPresheafedSpace.map g.val) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace f.val g.val).hom).base).op.obj
(op U) =
(Opens.map (SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π f.val g.val)).base).op.obj (op U)))) | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
/-!
# Colimits of LocallyRingedSpace
We construct the explicit coproducts and coequalizers of `LocallyRingedSpace`.
It then follows that `LocallyRingedSpace` has all colimits, and
`forget_to_SheafedSpace` preserves them.
-/
set_option linter.uppercaseLean3 false
namespace AlgebraicGeometry
universe v u
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
namespace SheafedSpace
variable {C : Type u} [Category.{v} C] [HasLimits C]
variable {J : Type v} [Category.{v} J] (F : J ⥤ SheafedSpace.{_, _, v} C)
theorem isColimit_exists_rep {c : Cocone F} (hc : IsColimit c) (x : c.pt) :
∃ (i : J) (y : F.obj i), (c.ι.app i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ forget C) (isColimitOfPreserves (forget C) hc) x
#align algebraic_geometry.SheafedSpace.is_colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.isColimit_exists_rep
-- Porting note : argument `C` of colimit need to be made explicit, odd
theorem colimit_exists_rep (x : colimit (C := SheafedSpace C) F) :
∃ (i : J) (y : F.obj i), (colimit.ι F i).base y = x :=
Concrete.isColimit_exists_rep (F ⋙ SheafedSpace.forget C)
(isColimitOfPreserves (SheafedSpace.forget _) (colimit.isColimit F)) x
#align algebraic_geometry.SheafedSpace.colimit_exists_rep AlgebraicGeometry.SheafedSpaceₓ.colimit_exists_rep
instance {X Y : SheafedSpace C} (f g : X ⟶ Y) : Epi (coequalizer.π f g).base := by
erw [←
show _ = (coequalizer.π f g).base from
ι_comp_coequalizerComparison f g (SheafedSpace.forget C)]
rw [← PreservesCoequalizer.iso_hom]
apply epi_comp
end SheafedSpace
namespace LocallyRingedSpace
section HasCoproducts
variable {ι : Type u} (F : Discrete ι ⥤ LocallyRingedSpace.{u})
-- Porting note : in this section, I marked `CommRingCat` as `CommRingCatMax.{u,u}`
-- This is a hack to avoid the following:
/-
```
stuck at solving universe constraint
u =?= max u ?u.11876
while trying to unify
HasLimits CommRingCat
with
(HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax) (HasLimitsOfSize CommRingCatMax)
```
-/
/-- The explicit coproduct for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproduct : LocallyRingedSpace where
toSheafedSpace := colimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace)
localRing x := by
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
haveI : LocalRing (((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace.stalk y) :=
(F.obj i).localRing _
exact
(asIso (PresheafedSpace.stalkMap
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i : _) y)).symm.commRingCatIsoToRingEquiv.localRing
#align algebraic_geometry.LocallyRingedSpace.coproduct AlgebraicGeometry.LocallyRingedSpace.coproduct
/-- The explicit coproduct cofan for `F : discrete ι ⥤ LocallyRingedSpace`. -/
noncomputable def coproductCofan : Cocone F where
pt := coproduct F
ι :=
{ app := fun j => ⟨colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) j, inferInstance⟩
naturality := fun ⟨j⟩ ⟨j'⟩ ⟨⟨(f : j = j')⟩⟩ => by subst f; aesop }
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan AlgebraicGeometry.LocallyRingedSpace.coproductCofan
/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/
noncomputable def coproductCofanIsColimit : IsColimit (coproductCofan F) where
desc s :=
⟨colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s), by
intro x
obtain ⟨i, y, ⟨⟩⟩ := SheafedSpace.colimit_exists_rep (F ⋙ forgetToSheafedSpace) x
have := PresheafedSpace.stalkMap.comp
(colimit.ι (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) i)
(colimit.desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(F ⋙ forgetToSheafedSpace) (forgetToSheafedSpace.mapCocone s)) y
rw [← IsIso.comp_inv_eq] at this
erw [← this,
PresheafedSpace.stalkMap.congr_hom _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u})
(forgetToSheafedSpace.mapCocone s) i : _)]
haveI :
IsLocalRingHom
(PresheafedSpace.stalkMap ((forgetToSheafedSpace.mapCocone s).ι.app i) y) :=
(s.ι.app i).2 y
infer_instance⟩
fac s j := LocallyRingedSpace.Hom.ext _ _
(colimit.ι_desc (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _ _)
uniq s f h :=
LocallyRingedSpace.Hom.ext _ _
(IsColimit.uniq _ (forgetToSheafedSpace.mapCocone s) f.1 fun j =>
congr_arg LocallyRingedSpace.Hom.val (h j))
#align algebraic_geometry.LocallyRingedSpace.coproduct_cofan_is_colimit AlgebraicGeometry.LocallyRingedSpace.coproductCofanIsColimit
instance : HasCoproducts.{u} LocallyRingedSpace.{u} := fun _ =>
⟨fun F => ⟨⟨⟨_, coproductCofanIsColimit F⟩⟩⟩⟩
noncomputable instance (J : Type _) :
PreservesColimitsOfShape (Discrete.{u} J) forgetToSheafedSpace.{u} :=
⟨fun {G} =>
preservesColimitOfPreservesColimitCocone (coproductCofanIsColimit G)
((colimit.isColimit (C := SheafedSpace.{u+1, u, u} CommRingCatMax.{u, u}) _).ofIsoColimit
(Cocones.ext (Iso.refl _) fun _ => Category.comp_id _))⟩
end HasCoproducts
section HasCoequalizer
variable {X Y : LocallyRingedSpace.{v}} (f g : X ⟶ Y)
namespace HasCoequalizer
instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
| rw [PresheafedSpace.comp_c_app, ← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π] | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) := by
have := ι_comp_coequalizerComparison f.1 g.1 SheafedSpace.forgetToPresheafedSpace
rw [← PreservesCoequalizer.iso_hom] at this
erw [SheafedSpace.congr_app this.symm (op U)]
| Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits.145_0.tE6q65npbp8AX2g | instance coequalizer_π_app_isLocalRingHom
(U : TopologicalSpace.Opens (coequalizer f.val g.val).carrier) :
IsLocalRingHom ((coequalizer.π f.val g.val : _).c.app (op U)) | Mathlib_Geometry_RingedSpace_LocallyRingedSpace_HasColimits |
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