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case h.e'_4
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
f g : ΞΉ β R
s t : Finset ΞΉ
h0 : β i β s, 0 β€ f i
h1 : β i β s, f i β€ 1
β’ β i in s, 1 = 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
| exact Finset.prod_const_one | /-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
| Mathlib.Algebra.BigOperators.Order.629_0.ewL52iF1Dz3xeLh | /-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i in s, g i + β i in s, h i β€ β i in s, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
| simp_rw [prod_eq_mul_prod_diff_singleton hi] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ f i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
| refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_) | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ (g i + h i) * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· | rw [right_distrib] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ g i * β i in s \ {i}, f i + h i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
| refine add_le_add ?_ ?_ | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ g i * β i in s \ {i}, g i β€ g i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· | refine mul_le_mul_of_nonneg_left ?_ ?_ | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i in s \ {i}, g i β€ β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· | refine prod_le_prod ?_ ?_ | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_1.refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s \ {i}, 0 β€ g i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | simp (config := { contextual := true }) [*] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_1.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s \ {i}, g i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | simp (config := { contextual := true }) [*] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ 0 β€ g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· | try apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ 0 β€ g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try | apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_2.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ i β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
| try assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_1.refine_2.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ i β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try | assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ h i * β i in s \ {i}, h i β€ h i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· | refine mul_le_mul_of_nonneg_left ?_ ?_ | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i in s \ {i}, h i β€ β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· | refine prod_le_prod ?_ ?_ | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_1.refine_1
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s \ {i}, 0 β€ h i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | simp (config := { contextual := true }) [*] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_1.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s \ {i}, h i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | simp (config := { contextual := true }) [*] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ 0 β€ h i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· | try apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ 0 β€ h i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try | apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_2.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ i β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
| try assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_1.refine_2.refine_2.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ i β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try | assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_2
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ 0 β€ β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· | apply prod_nonneg | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· | Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_2.h0
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s \ {i}, 0 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
| simp only [and_imp, mem_sdiff, mem_singleton] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_2.h0
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
β’ β i_1 β s, Β¬i_1 = i β 0 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
| intro j h1j h2j | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case refine_2.h0
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : OrderedCommSemiring R
fβ gβ : ΞΉ β R
s t : Finset ΞΉ
i : ΞΉ
f g h : ΞΉ β R
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
hg : β i β s, 0 β€ g i
hh : β i β s, 0 β€ h i
j : ΞΉ
h1j : j β s
h2j : Β¬j = i
β’ 0 β€ f j | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
| exact le_trans (hg j h1j) (hgf j h1j h2j) | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
| Mathlib.Algebra.BigOperators.Order.636_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
hlt : β i β s, f i < g i
β’ β i in s, f i < β i in s, g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
| classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
| Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
hlt : β i β s, f i < g i
β’ β i in s, f i < β i in s, g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
| obtain β¨i, hi, hiltβ© := hlt | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
| Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro.intro
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
i : ΞΉ
hi : i β s
hilt : f i < g i
β’ β i in s, f i < β i in s, g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
| rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)] | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
| Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro.intro
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
i : ΞΉ
hi : i β s
hilt : f i < g i
β’ f i * β x in erase s i, f x < g i * β x in erase s i, g x | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
| apply mul_lt_mul hilt | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
| Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro.intro.hbd
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
i : ΞΉ
hi : i β s
hilt : f i < g i
β’ β x in erase s i, f x β€ β x in erase s i, g x | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· | exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj) | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· | Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro.intro.hb
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
i : ΞΉ
hi : i β s
hilt : f i < g i
β’ 0 < β x in erase s i, f x | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· | exact prod_pos fun j hj => hf j (mem_of_mem_erase hj) | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· | Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro.intro.hc
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i β€ g i
i : ΞΉ
hi : i β s
hilt : f i < g i
β’ 0 β€ g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· | exact le_of_lt <| (hf i hi).trans hilt | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· | Mathlib.Algebra.BigOperators.Order.667_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i < g i
h_ne : Finset.Nonempty s
β’ β i in s, f i < β i in s, g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
| apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi) | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
| Mathlib.Algebra.BigOperators.Order.679_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i < g i
h_ne : Finset.Nonempty s
β’ β i β s, f i < g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
| obtain β¨i, hiβ© := h_ne | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
| Mathlib.Algebra.BigOperators.Order.679_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
case intro
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : StrictOrderedCommSemiring R
f g : ΞΉ β R
s : Finset ΞΉ
hf : β i β s, 0 < f i
hfg : β i β s, f i < g i
i : ΞΉ
hi : i β s
β’ β i β s, f i < g i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
| exact β¨i, hi, hfg i hiβ© | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
| Mathlib.Algebra.BigOperators.Order.679_0.ewL52iF1Dz3xeLh | theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i in s, g i + β i in s, h i β€ β i in s, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
| classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i in s, g i + β i in s, h i β€ β i in s, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
| simp_rw [prod_eq_mul_prod_diff_singleton hi] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ f i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
| refine' le_trans _ (mul_le_mul_right' h2i _) | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ (g i + h i) * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
| rw [right_distrib] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ g i * β i in s \ {i}, g i + h i * β i in s \ {i}, h i β€ g i * β i in s \ {i}, f i + h i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
| apply add_le_add | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ g i * β i in s \ {i}, g i β€ g i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> | apply mul_le_mul_left' | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> | Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ h i * β i in s \ {i}, h i β€ h i * β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> | apply mul_le_mul_left' | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> | Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i in s \ {i}, g i β€ β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> | apply prod_le_prod' | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> | Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i in s \ {i}, h i β€ β i in s \ {i}, f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> | apply prod_le_prod' | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> | Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i_1 β s \ {i}, g i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
| simp only [and_imp, mem_sdiff, mem_singleton] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i_1 β s \ {i}, h i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
| simp only [and_imp, mem_sdiff, mem_singleton] | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i_1 β s, Β¬i_1 = i β g i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
| intros | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
β’ β i_1 β s, Β¬i_1 = i β h i_1 β€ f i_1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
| intros | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ g iβ β€ f iβ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
| apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ h iβ β€ f iβ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
| apply_assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ iβ β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ iβ β i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ iβ β s | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
case hβ.bc.h.a
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : CanonicallyOrderedCommSemiring R
f g h : ΞΉ β R
s : Finset ΞΉ
i : ΞΉ
hi : i β s
h2i : g i + h i β€ f i
hgf : β j β s, j β i β g j β€ f j
hhf : β j β s, j β i β h j β€ f j
iβ : ΞΉ
aβΒΉ : iβ β s
aβ : Β¬iβ = i
β’ iβ β i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| assumption | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
| Mathlib.Algebra.BigOperators.Order.698_0.ewL52iF1Dz3xeLh | /-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : 1 < f
β’ β i β Finset.univ, 1 < f i | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by | simpa using (Pi.lt_def.1 hf).2 | @[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by | Mathlib.Algebra.BigOperators.Order.746_0.ewL52iF1Dz3xeLh | @[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : f < 1
β’ β i β Finset.univ, f i < 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by | simpa using (Pi.lt_def.1 hf).2 | @[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by | Mathlib.Algebra.BigOperators.Order.750_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : 1 β€ f
β’ 1 < β i : ΞΉ, f i β 1 < f | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
| obtain rfl | hf := hf.eq_or_lt | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
| Mathlib.Algebra.BigOperators.Order.754_0.ewL52iF1Dz3xeLh | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f | Mathlib_Algebra_BigOperators_Order |
case inl
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
hf : 1 β€ 1
β’ 1 < β i : ΞΉ, OfNat.ofNat 1 i β 1 < 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> | simp [*, one_lt_prod] | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> | Mathlib.Algebra.BigOperators.Order.754_0.ewL52iF1Dz3xeLh | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f | Mathlib_Algebra_BigOperators_Order |
case inr
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hfβ : 1 β€ f
hf : 1 < f
β’ 1 < β i : ΞΉ, f i β 1 < f | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> | simp [*, one_lt_prod] | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> | Mathlib.Algebra.BigOperators.Order.754_0.ewL52iF1Dz3xeLh | @[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : f β€ 1
β’ β i : ΞΉ, f i < 1 β f < 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
| obtain rfl | hf := hf.eq_or_lt | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
| Mathlib.Algebra.BigOperators.Order.758_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 | Mathlib_Algebra_BigOperators_Order |
case inl
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
hf : 1 β€ 1
β’ β i : ΞΉ, OfNat.ofNat 1 i < 1 β 1 < 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> | simp [*, prod_lt_one] | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> | Mathlib.Algebra.BigOperators.Order.758_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 | Mathlib_Algebra_BigOperators_Order |
case inr
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hfβ : f β€ 1
hf : f < 1
β’ β i : ΞΉ, f i < 1 β f < 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> | simp [*, prod_lt_one] | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> | Mathlib.Algebra.BigOperators.Order.758_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : 1 β€ f
β’ β i : ΞΉ, f i = 1 β f = 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
| simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not | @[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
| Mathlib.Algebra.BigOperators.Order.762_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : Fintype ΞΉ
instβ : OrderedCancelCommMonoid M
f : ΞΉ β M
hf : f β€ 1
β’ β i : ΞΉ, f i = 1 β f = 1 | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not
@[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
| simpa only [(prod_le_one hf).not_gt_iff_eq, hf.not_gt_iff_eq, eq_comm]
using (prod_lt_one_iff_of_le_one hf).not | @[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
| Mathlib.Algebra.BigOperators.Order.767_0.ewL52iF1Dz3xeLh | @[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : AddCommMonoid M
s : Finset ΞΉ
f : ΞΉ β WithTop M
β’ β i in s, f i = β€ β β i β s, f i = β€ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not
@[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
simpa only [(prod_le_one hf).not_gt_iff_eq, hf.not_gt_iff_eq, eq_comm]
using (prod_lt_one_iff_of_le_one hf).not
end OrderedCancelCommMonoid
end Fintype
namespace WithTop
open Finset
/-- A product of finite numbers is still finite -/
theorem prod_lt_top [CommMonoidWithZero R] [NoZeroDivisors R] [Nontrivial R] [DecidableEq R] [LT R]
{s : Finset ΞΉ} {f : ΞΉ β WithTop R} (h : β i β s, f i β β€) : β i in s, f i < β€ :=
prod_induction f (fun a β¦ a < β€) (fun _ _ hβ hβ β¦ mul_lt_top' hβ hβ) (coe_lt_top 1)
fun a ha β¦ WithTop.lt_top_iff_ne_top.2 (h a ha)
#align with_top.prod_lt_top WithTop.prod_lt_top
/-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
| induction s using Finset.cons_induction | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
| Mathlib.Algebra.BigOperators.Order.786_0.ewL52iF1Dz3xeLh | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ | Mathlib_Algebra_BigOperators_Order |
case empty
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : AddCommMonoid M
f : ΞΉ β WithTop M
β’ β i in β
, f i = β€ β β i β β
, f i = β€ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not
@[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
simpa only [(prod_le_one hf).not_gt_iff_eq, hf.not_gt_iff_eq, eq_comm]
using (prod_lt_one_iff_of_le_one hf).not
end OrderedCancelCommMonoid
end Fintype
namespace WithTop
open Finset
/-- A product of finite numbers is still finite -/
theorem prod_lt_top [CommMonoidWithZero R] [NoZeroDivisors R] [Nontrivial R] [DecidableEq R] [LT R]
{s : Finset ΞΉ} {f : ΞΉ β WithTop R} (h : β i β s, f i β β€) : β i in s, f i < β€ :=
prod_induction f (fun a β¦ a < β€) (fun _ _ hβ hβ β¦ mul_lt_top' hβ hβ) (coe_lt_top 1)
fun a ha β¦ WithTop.lt_top_iff_ne_top.2 (h a ha)
#align with_top.prod_lt_top WithTop.prod_lt_top
/-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
induction s using Finset.cons_induction <;> | simp [*] | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
induction s using Finset.cons_induction <;> | Mathlib.Algebra.BigOperators.Order.786_0.ewL52iF1Dz3xeLh | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ | Mathlib_Algebra_BigOperators_Order |
case cons
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβ : AddCommMonoid M
f : ΞΉ β WithTop M
aβΒΉ : ΞΉ
sβ : Finset ΞΉ
hβ : aβΒΉ β sβ
aβ : β i in sβ, f i = β€ β β i β sβ, f i = β€
β’ β i in cons aβΒΉ sβ hβ, f i = β€ β β i β cons aβΒΉ sβ hβ, f i = β€ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not
@[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
simpa only [(prod_le_one hf).not_gt_iff_eq, hf.not_gt_iff_eq, eq_comm]
using (prod_lt_one_iff_of_le_one hf).not
end OrderedCancelCommMonoid
end Fintype
namespace WithTop
open Finset
/-- A product of finite numbers is still finite -/
theorem prod_lt_top [CommMonoidWithZero R] [NoZeroDivisors R] [Nontrivial R] [DecidableEq R] [LT R]
{s : Finset ΞΉ} {f : ΞΉ β WithTop R} (h : β i β s, f i β β€) : β i in s, f i < β€ :=
prod_induction f (fun a β¦ a < β€) (fun _ _ hβ hβ β¦ mul_lt_top' hβ hβ) (coe_lt_top 1)
fun a ha β¦ WithTop.lt_top_iff_ne_top.2 (h a ha)
#align with_top.prod_lt_top WithTop.prod_lt_top
/-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
induction s using Finset.cons_induction <;> | simp [*] | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
induction s using Finset.cons_induction <;> | Mathlib.Algebra.BigOperators.Order.786_0.ewL52iF1Dz3xeLh | /-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ | Mathlib_Algebra_BigOperators_Order |
ΞΉ : Type u_1
Ξ± : Type u_2
Ξ² : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
instβΒΉ : AddCommMonoid M
instβ : LT M
s : Finset ΞΉ
f : ΞΉ β WithTop M
β’ β i in s, f i < β€ β β i β s, f i < β€ | /-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.BigOperators.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Tactic.GCongr.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
/-!
# Results about big operators with values in an ordered algebraic structure.
Mostly monotonicity results for the `β` and `β` operations.
-/
open Function
open BigOperators
variable {ΞΉ Ξ± Ξ² M N G k R : Type*}
namespace Finset
section OrderedCommMonoid
variable [CommMonoid M] [OrderedCommMonoid N]
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
submultiplicative on `{x | p x}`, i.e., `p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be
a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
@[to_additive le_sum_nonempty_of_subadditive_on_pred]
theorem le_prod_nonempty_of_submultiplicative_on_pred (f : M β N) (p : M β Prop)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) (s : Finset ΞΉ) (hs_nonempty : s.Nonempty) (hs : β i β s, p (g i)) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _
Β· simp [hs_nonempty.ne_empty]
Β· exact Multiset.forall_mem_map_iff.mpr hs
rw [Multiset.map_map]
rfl
#align finset.le_prod_nonempty_of_submultiplicative_on_pred Finset.le_prod_nonempty_of_submultiplicative_on_pred
#align finset.le_sum_nonempty_of_subadditive_on_pred Finset.le_sum_nonempty_of_subadditive_on_pred
/-- Let `{x | p x}` be an additive subsemigroup of an additive commutative monoid `M`. Let
`f : M β N` be a map subadditive on `{x | p x}`, i.e., `p x β p y β f (x + y) β€ f x + f y`. Let
`g i`, `i β s`, be a nonempty finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_nonempty_of_subadditive]
theorem le_prod_nonempty_of_submultiplicative (f : M β N) (h_mul : β x y, f (x * y) β€ f x * f y)
{s : Finset ΞΉ} (hs : s.Nonempty) (g : ΞΉ β M) : f (β i in s, g i) β€ β i in s, f (g i) :=
le_prod_nonempty_of_submultiplicative_on_pred f (fun _ β¦ True) (fun x y _ _ β¦ h_mul x y)
(fun _ _ _ _ β¦ trivial) g s hs fun _ _ β¦ trivial
#align finset.le_prod_nonempty_of_submultiplicative Finset.le_prod_nonempty_of_submultiplicative
#align finset.le_sum_nonempty_of_subadditive Finset.le_sum_nonempty_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y` and `g i`, `i β s`, is a
nonempty finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_nonempty_of_subadditive
/-- Let `{x | p x}` be a subsemigroup of a commutative monoid `M`. Let `f : M β N` be a map
such that `f 1 = 1` and `f` is submultiplicative on `{x | p x}`, i.e.,
`p x β p y β f (x * y) β€ f x * f y`. Let `g i`, `i β s`, be a finite family of elements of `M` such
that `β i β s, p (g i)`. Then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive_on_pred]
theorem le_prod_of_submultiplicative_on_pred (f : M β N) (p : M β Prop) (h_one : f 1 = 1)
(h_mul : β x y, p x β p y β f (x * y) β€ f x * f y) (hp_mul : β x y, p x β p y β p (x * y))
(g : ΞΉ β M) {s : Finset ΞΉ} (hs : β i β s, p (g i)) : f (β i in s, g i) β€ β i in s, f (g i) := by
rcases eq_empty_or_nonempty s with (rfl | hs_nonempty)
Β· simp [h_one]
Β· exact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs
#align finset.le_prod_of_submultiplicative_on_pred Finset.le_prod_of_submultiplicative_on_pred
#align finset.le_sum_of_subadditive_on_pred Finset.le_sum_of_subadditive_on_pred
/-- Let `{x | p x}` be a subsemigroup of a commutative additive monoid `M`. Let `f : M β N` be a map
such that `f 0 = 0` and `f` is subadditive on `{x | p x}`, i.e. `p x β p y β f (x + y) β€ f x + f y`.
Let `g i`, `i β s`, be a finite family of elements of `M` such that `β i β s, p (g i)`. Then
`f (β x in s, g x) β€ β x in s, f (g x)`. -/
add_decl_doc le_sum_of_subadditive_on_pred
/-- If `f : M β N` is a submultiplicative function, `f (x * y) β€ f x * f y`, `f 1 = 1`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
@[to_additive le_sum_of_subadditive]
theorem le_prod_of_submultiplicative (f : M β N) (h_one : f 1 = 1)
(h_mul : β x y, f (x * y) β€ f x * f y) (s : Finset ΞΉ) (g : ΞΉ β M) :
f (β i in s, g i) β€ β i in s, f (g i) := by
refine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _
rw [Multiset.map_map]
rfl
#align finset.le_prod_of_submultiplicative Finset.le_prod_of_submultiplicative
#align finset.le_sum_of_subadditive Finset.le_sum_of_subadditive
/-- If `f : M β N` is a subadditive function, `f (x + y) β€ f x + f y`, `f 0 = 0`, and `g i`,
`i β s`, is a finite family of elements of `M`, then `f (β i in s, g i) β€ β i in s, f (g i)`. -/
add_decl_doc le_sum_of_subadditive
variable {f g : ΞΉ β N} {s t : Finset ΞΉ}
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then
`β i in s, f i β€ β i in s, g i`. -/
@[to_additive sum_le_sum]
theorem prod_le_prod' (h : β i β s, f i β€ g i) : β i in s, f i β€ β i in s, g i :=
Multiset.prod_map_le_prod_map f g h
#align finset.prod_le_prod' Finset.prod_le_prod'
#align finset.sum_le_sum Finset.sum_le_sum
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then
`β i in s, f i β€ β i in s, g i`. -/
add_decl_doc sum_le_sum
/-- In an ordered commutative monoid, if each factor `f i` of one finite product is less than or
equal to the corresponding factor `g i` of another finite product, then `s.prod f β€ s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod'`, convenient
for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_le_sum]
theorem _root_.GCongr.prod_le_prod' (h : β i β s, f i β€ g i) : s.prod f β€ s.prod g :=
s.prod_le_prod' h
/-- In an ordered additive commutative monoid, if each summand `f i` of one finite sum is less than
or equal to the corresponding summand `g i` of another finite sum, then `s.sum f β€ s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_le_sum`, convenient
for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_le_sum
@[to_additive sum_nonneg]
theorem one_le_prod' (h : β i β s, 1 β€ f i) : 1 β€ β i in s, f i :=
le_trans (by rw [prod_const_one]) (prod_le_prod' h)
#align finset.one_le_prod' Finset.one_le_prod'
#align finset.sum_nonneg Finset.sum_nonneg
@[to_additive Finset.sum_nonneg']
theorem one_le_prod'' (h : β i : ΞΉ, 1 β€ f i) : 1 β€ β i : ΞΉ in s, f i :=
Finset.one_le_prod' fun i _ β¦ h i
#align finset.one_le_prod'' Finset.one_le_prod''
#align finset.sum_nonneg' Finset.sum_nonneg'
@[to_additive sum_nonpos]
theorem prod_le_one' (h : β i β s, f i β€ 1) : β i in s, f i β€ 1 :=
(prod_le_prod' h).trans_eq (by rw [prod_const_one])
#align finset.prod_le_one' Finset.prod_le_one'
#align finset.sum_nonpos Finset.sum_nonpos
@[to_additive sum_le_sum_of_subset_of_nonneg]
theorem prod_le_prod_of_subset_of_one_le' (h : s β t) (hf : β i β t, i β s β 1 β€ f i) :
β i in s, f i β€ β i in t, f i := by
classical calc
β i in s, f i β€ (β i in t \ s, f i) * β i in s, f i :=
le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]
_ = β i in t \ s βͺ s, f i := (prod_union sdiff_disjoint).symm
_ = β i in t, f i := by rw [sdiff_union_of_subset h]
#align finset.prod_le_prod_of_subset_of_one_le' Finset.prod_le_prod_of_subset_of_one_le'
#align finset.sum_le_sum_of_subset_of_nonneg Finset.sum_le_sum_of_subset_of_nonneg
@[to_additive sum_mono_set_of_nonneg]
theorem prod_mono_set_of_one_le' (hf : β x, 1 β€ f x) : Monotone fun s β¦ β x in s, f x :=
fun _ _ hst β¦ prod_le_prod_of_subset_of_one_le' hst fun x _ _ β¦ hf x
#align finset.prod_mono_set_of_one_le' Finset.prod_mono_set_of_one_le'
#align finset.sum_mono_set_of_nonneg Finset.sum_mono_set_of_nonneg
@[to_additive sum_le_univ_sum_of_nonneg]
theorem prod_le_univ_prod_of_one_le' [Fintype ΞΉ] {s : Finset ΞΉ} (w : β x, 1 β€ f x) :
β x in s, f x β€ β x, f x :=
prod_le_prod_of_subset_of_one_le' (subset_univ s) fun a _ _ β¦ w a
#align finset.prod_le_univ_prod_of_one_le' Finset.prod_le_univ_prod_of_one_le'
#align finset.sum_le_univ_sum_of_nonneg Finset.sum_le_univ_sum_of_nonneg
-- Porting Note: TODO -- The two next lemmas give the same lemma in additive version
@[to_additive sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_one_le' :
(β i β s, 1 β€ f i) β ((β i in s, f i) = 1 β β i β s, f i = 1) := by
classical
refine Finset.induction_on s
(fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h), fun _ β¦ rflβ©) ?_
intro a s ha ih H
have : β i β s, 1 β€ f i := fun _ β¦ H _ β mem_insert_of_mem
rw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this),
forall_mem_insert, ih this]
#align finset.prod_eq_one_iff_of_one_le' Finset.prod_eq_one_iff_of_one_le'
#align finset.sum_eq_zero_iff_of_nonneg Finset.sum_eq_zero_iff_of_nonneg
@[to_additive existing sum_eq_zero_iff_of_nonneg]
theorem prod_eq_one_iff_of_le_one' :
(β i β s, f i β€ 1) β ((β i in s, f i) = 1 β β i β s, f i = 1) :=
@prod_eq_one_iff_of_one_le' _ Nα΅α΅ _ _ _
#align finset.prod_eq_one_iff_of_le_one' Finset.prod_eq_one_iff_of_le_one'
-- Porting note: there is no align for the additive version since it aligns to the
-- same one as the previous lemma
@[to_additive single_le_sum]
theorem single_le_prod' (hf : β i β s, 1 β€ f i) {a} (h : a β s) : f a β€ β x in s, f x :=
calc
f a = β i in {a}, f i := (prod_singleton _ _).symm
_ β€ β i in s, f i :=
prod_le_prod_of_subset_of_one_le' (singleton_subset_iff.2 h) fun i hi _ β¦ hf i hi
#align finset.single_le_prod' Finset.single_le_prod'
#align finset.single_le_sum Finset.single_le_sum
@[to_additive]
lemma mul_le_prod {i j : ΞΉ} (hf : β i β s, 1 β€ f i) (hi : i β s) (hj : j β s) (hne : i β j) :
f i * f j β€ β k in s, f k :=
calc
f i * f j = β k in .cons i {j} (by simpa), f k := by rw [prod_cons, prod_singleton]
_ β€ β k in s, f k := by
refine prod_le_prod_of_subset_of_one_le' ?_ fun k hk _ β¦ hf k hk
simp [cons_subset, *]
@[to_additive sum_le_card_nsmul]
theorem prod_le_pow_card (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, f x β€ n) :
s.prod f β€ n ^ s.card := by
refine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _
Β· simpa using h
Β· simp
#align finset.prod_le_pow_card Finset.prod_le_pow_card
#align finset.sum_le_card_nsmul Finset.sum_le_card_nsmul
@[to_additive card_nsmul_le_sum]
theorem pow_card_le_prod (s : Finset ΞΉ) (f : ΞΉ β N) (n : N) (h : β x β s, n β€ f x) :
n ^ s.card β€ s.prod f := @Finset.prod_le_pow_card _ Nα΅α΅ _ _ _ _ h
#align finset.pow_card_le_prod Finset.pow_card_le_prod
#align finset.card_nsmul_le_sum Finset.card_nsmul_le_sum
theorem card_biUnion_le_card_mul [DecidableEq Ξ²] (s : Finset ΞΉ) (f : ΞΉ β Finset Ξ²) (n : β)
(h : β a β s, (f a).card β€ n) : (s.biUnion f).card β€ s.card * n :=
card_biUnion_le.trans <| sum_le_card_nsmul _ _ _ h
#align finset.card_bUnion_le_card_mul Finset.card_biUnion_le_card_mul
variable {ΞΉ' : Type*} [DecidableEq ΞΉ']
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_fiberwise_le_sum_of_sum_fiber_nonneg]
theorem prod_fiberwise_le_prod_of_one_le_prod_fiber' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, (1 : N) β€ β x in s.filter fun x β¦ g x = y, f x) :
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€ β x in s, f x :=
calc
(β y in t, β x in s.filter fun x β¦ g x = y, f x) β€
β y in t βͺ s.image g, β x in s.filter fun x β¦ g x = y, f x :=
prod_le_prod_of_subset_of_one_le' (subset_union_left _ _) fun y _ β¦ h y
_ = β x in s, f x :=
prod_fiberwise_of_maps_to (fun _ hx β¦ mem_union.2 <| Or.inr <| mem_image_of_mem _ hx) _
#align finset.prod_fiberwise_le_prod_of_one_le_prod_fiber' Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
#align finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg Finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg
-- Porting note: Mathport warning: expanding binder collection (y Β«expr β Β» t)
@[to_additive sum_le_sum_fiberwise_of_sum_fiber_nonpos]
theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ΞΉ'} {g : ΞΉ β ΞΉ'} {f : ΞΉ β N}
(h : β y β t, β x in s.filter fun x β¦ g x = y, f x β€ 1) :
β x in s, f x β€ β y in t, β x in s.filter fun x β¦ g x = y, f x :=
@prod_fiberwise_le_prod_of_one_le_prod_fiber' _ Nα΅α΅ _ _ _ _ _ _ _ h
#align finset.prod_le_prod_fiberwise_of_prod_fiber_le_one' Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
#align finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos Finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos
end OrderedCommMonoid
theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ΞΉ β G) (s : Finset ΞΉ) :
|β i in s, f i| β€ β i in s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
#align finset.abs_sum_le_sum_abs Finset.abs_sum_le_sum_abs
theorem abs_sum_of_nonneg {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i β s, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg hf)]
#align finset.abs_sum_of_nonneg Finset.abs_sum_of_nonneg
theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ΞΉ β G} {s : Finset ΞΉ}
(hf : β i, 0 β€ f i) : |β i : ΞΉ in s, f i| = β i : ΞΉ in s, f i := by
rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
#align finset.abs_sum_of_nonneg' Finset.abs_sum_of_nonneg'
theorem abs_prod {R : Type*} [LinearOrderedCommRing R] {f : ΞΉ β R} {s : Finset ΞΉ} :
|β x in s, f x| = β x in s, |f x| :=
map_prod absHom _ _
#align finset.abs_prod Finset.abs_prod
section Pigeonhole
variable [DecidableEq Ξ²]
theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a in t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a in t, n := sum_le_sum hn
_ = _ := by simp [mul_comm]
#align finset.card_le_mul_card_image_of_maps_to Finset.card_le_mul_card_image_of_maps_to
theorem card_le_mul_card_image {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, (s.filter fun x β¦ f x = a).card β€ n) : s.card β€ n * (s.image f).card :=
card_le_mul_card_image_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.card_le_mul_card_image Finset.card_le_mul_card_image
theorem mul_card_image_le_card_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, n β€ (s.filter fun x β¦ f x = a).card) :
n * t.card β€ s.card :=
calc
n * t.card = β _a in t, n := by simp [mul_comm]
_ β€ β a in t, (s.filter fun x β¦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [β card_eq_sum_card_fiberwise Hf]
#align finset.mul_card_image_le_card_of_maps_to Finset.mul_card_image_le_card_of_maps_to
theorem mul_card_image_le_card {f : Ξ± β Ξ²} (s : Finset Ξ±) (n : β)
(hn : β a β s.image f, n β€ (s.filter fun x β¦ f x = a).card) : n * (s.image f).card β€ s.card :=
mul_card_image_le_card_of_maps_to (fun _ β¦ mem_image_of_mem _) n hn
#align finset.mul_card_image_le_card Finset.mul_card_image_le_card
end Pigeonhole
section DoubleCounting
variable [DecidableEq Ξ±] {s : Finset Ξ±} {B : Finset (Finset Ξ±)} {n : β}
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_inter_le (h : β a β s, (B.filter (a β Β·)).card β€ n) :
(β t in B, (s β© t).card) β€ s.card * n := by
refine' le_trans _ (s.sum_le_card_nsmul _ _ h)
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.sum_card_inter_le Finset.sum_card_inter_le
/-- If every element belongs to at most `n` Finsets, then the sum of their sizes is at most `n`
times how many they are. -/
theorem sum_card_le [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card β€ n) :
β s in B, s.card β€ Fintype.card Ξ± * n :=
calc
β s in B, s.card = β s in B, (univ β© s).card := by simp_rw [univ_inter]
_ β€ Fintype.card Ξ± * n := sum_card_inter_le fun a _ β¦ h a
#align finset.sum_card_le Finset.sum_card_le
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card_inter (h : β a β s, n β€ (B.filter (a β Β·)).card) :
s.card * n β€ β t in B, (s β© t).card := by
apply (s.card_nsmul_le_sum _ _ h).trans
simp_rw [β filter_mem_eq_inter, card_eq_sum_ones, sum_filter]
exact sum_comm.le
#align finset.le_sum_card_inter Finset.le_sum_card_inter
/-- If every element belongs to at least `n` Finsets, then the sum of their sizes is at least `n`
times how many they are. -/
theorem le_sum_card [Fintype Ξ±] (h : β a, n β€ (B.filter (a β Β·)).card) :
Fintype.card Ξ± * n β€ β s in B, s.card :=
calc
Fintype.card Ξ± * n β€ β s in B, (univ β© s).card := le_sum_card_inter fun a _ β¦ h a
_ = β s in B, s.card := by simp_rw [univ_inter]
#align finset.le_sum_card Finset.le_sum_card
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card_inter (h : β a β s, (B.filter (a β Β·)).card = n) :
(β t in B, (s β© t).card) = s.card * n :=
(sum_card_inter_le fun a ha β¦ (h a ha).le).antisymm (le_sum_card_inter fun a ha β¦ (h a ha).ge)
#align finset.sum_card_inter Finset.sum_card_inter
/-- If every element belongs to exactly `n` Finsets, then the sum of their sizes is `n` times how
many they are. -/
theorem sum_card [Fintype Ξ±] (h : β a, (B.filter (a β Β·)).card = n) :
β s in B, s.card = Fintype.card Ξ± * n := by
simp_rw [Fintype.card, β sum_card_inter fun a _ β¦ h a, univ_inter]
#align finset.sum_card Finset.sum_card
theorem card_le_card_biUnion {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hs : (s : Set ΞΉ).PairwiseDisjoint f)
(hf : β i β s, (f i).Nonempty) : s.card β€ (s.biUnion f).card := by
rw [card_biUnion hs, card_eq_sum_ones]
exact sum_le_sum fun i hi β¦ (hf i hi).card_pos
#align finset.card_le_card_bUnion Finset.card_le_card_biUnion
theorem card_le_card_biUnion_add_card_fiber {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±}
(hs : (s : Set ΞΉ).PairwiseDisjoint f) :
s.card β€ (s.biUnion f).card + (s.filter fun i β¦ f i = β
).card := by
rw [β Finset.filter_card_add_filter_neg_card_eq_card fun i β¦ f i = β
, add_comm]
exact
add_le_add_right
((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi β¦
nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|
card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)
_
#align finset.card_le_card_bUnion_add_card_fiber Finset.card_le_card_biUnion_add_card_fiber
theorem card_le_card_biUnion_add_one {s : Finset ΞΉ} {f : ΞΉ β Finset Ξ±} (hf : Injective f)
(hs : (s : Set ΞΉ).PairwiseDisjoint f) : s.card β€ (s.biUnion f).card + 1 :=
(card_le_card_biUnion_add_card_fiber hs).trans <|
add_le_add_left
(card_le_one.2 fun _ hi _ hj β¦ hf <| (mem_filter.1 hi).2.trans (mem_filter.1 hj).2.symm) _
#align finset.card_le_card_bUnion_add_one Finset.card_le_card_biUnion_add_one
end DoubleCounting
section CanonicallyOrderedCommMonoid
variable [CanonicallyOrderedCommMonoid M] {f : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive (attr := simp) sum_eq_zero_iff]
theorem prod_eq_one_iff' : β x in s, f x = 1 β β x β s, f x = 1 :=
prod_eq_one_iff_of_one_le' fun x _ β¦ one_le (f x)
#align finset.prod_eq_one_iff' Finset.prod_eq_one_iff'
#align finset.sum_eq_zero_iff Finset.sum_eq_zero_iff
@[to_additive sum_le_sum_of_subset]
theorem prod_le_prod_of_subset' (h : s β t) : β x in s, f x β€ β x in t, f x :=
prod_le_prod_of_subset_of_one_le' h fun _ _ _ β¦ one_le _
#align finset.prod_le_prod_of_subset' Finset.prod_le_prod_of_subset'
#align finset.sum_le_sum_of_subset Finset.sum_le_sum_of_subset
@[to_additive sum_mono_set]
theorem prod_mono_set' (f : ΞΉ β M) : Monotone fun s β¦ β x in s, f x := fun _ _ hs β¦
prod_le_prod_of_subset' hs
#align finset.prod_mono_set' Finset.prod_mono_set'
#align finset.sum_mono_set Finset.sum_mono_set
@[to_additive sum_le_sum_of_ne_zero]
theorem prod_le_prod_of_ne_one' (h : β x β s, f x β 1 β x β t) :
β x in s, f x β€ β x in t, f x := by
classical calc
β x in s, f x = (β x in s.filter fun x β¦ f x = 1, f x) *
β x in s.filter fun x β¦ f x β 1, f x := by
rw [β prod_union, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ h n_h β¦ n_h h
_ β€ β x in t, f x :=
mul_le_of_le_one_of_le
(prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ β¦ le_of_eq)
(prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp] )
#align finset.prod_le_prod_of_ne_one' Finset.prod_le_prod_of_ne_one'
#align finset.sum_le_sum_of_ne_zero Finset.sum_le_sum_of_ne_zero
end CanonicallyOrderedCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive sum_lt_sum]
theorem prod_lt_prod' (hle : β i β s, f i β€ g i) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod' hle hlt
#align finset.prod_lt_prod' Finset.prod_lt_prod'
#align finset.sum_lt_sum Finset.sum_lt_sum
@[to_additive sum_lt_sum_of_nonempty]
theorem prod_lt_prod_of_nonempty' (hs : s.Nonempty) (hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i :=
Multiset.prod_lt_prod_of_nonempty' (by aesop) hlt
#align finset.prod_lt_prod_of_nonempty' Finset.prod_lt_prod_of_nonempty'
#align finset.sum_lt_sum_of_nonempty Finset.sum_lt_sum_of_nonempty
/-- In an ordered commutative monoid, if each factor `f i` of one nontrivial finite product is
strictly less than the corresponding factor `g i` of another nontrivial finite product, then
`s.prod f < s.prod g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_lt_prod_of_nonempty'`,
convenient for the `gcongr` tactic. -/
@[to_additive (attr := gcongr) GCongr.sum_lt_sum_of_nonempty]
theorem _root_.GCongr.prod_lt_prod_of_nonempty' (hs : s.Nonempty) (Hlt : β i β s, f i < g i) :
s.prod f < s.prod g :=
s.prod_lt_prod_of_nonempty' hs Hlt
/-- In an ordered additive commutative monoid, if each summand `f i` of one nontrivial finite sum is
strictly less than the corresponding summand `g i` of another nontrivial finite sum, then
`s.sum f < s.sum g`.
This is a variant (beta-reduced) version of the standard lemma `Finset.sum_lt_sum_of_nonempty`,
convenient for the `gcongr` tactic. -/
add_decl_doc GCongr.sum_lt_sum_of_nonempty
-- Porting note: TODO -- calc indentation
@[to_additive sum_lt_sum_of_subset]
theorem prod_lt_prod_of_subset' (h : s β t) {i : ΞΉ} (ht : i β t) (hs : i β s) (hlt : 1 < f i)
(hle : β j β t, j β s β 1 β€ f j) : β j in s, f j < β j in t, f j := by
classical calc
β j in s, f j < β j in insert i s, f j := by
rw [prod_insert hs]
exact lt_mul_of_one_lt_left' (β j in s, f j) hlt
_ β€ β j in t, f j := by
apply prod_le_prod_of_subset_of_one_le'
Β· simp [Finset.insert_subset_iff, h, ht]
Β· intro x hx h'x
simp only [mem_insert, not_or] at h'x
exact hle x hx h'x.2
#align finset.prod_lt_prod_of_subset' Finset.prod_lt_prod_of_subset'
#align finset.sum_lt_sum_of_subset Finset.sum_lt_sum_of_subset
@[to_additive single_lt_sum]
theorem single_lt_prod' {i j : ΞΉ} (hij : j β i) (hi : i β s) (hj : j β s) (hlt : 1 < f j)
(hle : β k β s, k β i β 1 β€ f k) : f i < β k in s, f k :=
calc
f i = β k in {i}, f k := by rw [prod_singleton]
_ < β k in s, f k :=
prod_lt_prod_of_subset' (singleton_subset_iff.2 hi) hj (mt mem_singleton.1 hij) hlt
fun k hks hki β¦ hle k hks (mt mem_singleton.2 hki)
#align finset.single_lt_prod' Finset.single_lt_prod'
#align finset.single_lt_sum Finset.single_lt_sum
@[to_additive sum_pos]
theorem one_lt_prod (h : β i β s, 1 < f i) (hs : s.Nonempty) : 1 < β i in s, f i :=
lt_of_le_of_lt (by rw [prod_const_one]) <| prod_lt_prod_of_nonempty' hs h
#align finset.one_lt_prod Finset.one_lt_prod
#align finset.sum_pos Finset.sum_pos
@[to_additive]
theorem prod_lt_one (h : β i β s, f i < 1) (hs : s.Nonempty) : β i in s, f i < 1 :=
(prod_lt_prod_of_nonempty' hs h).trans_le (by rw [prod_const_one])
#align finset.prod_lt_one Finset.prod_lt_one
#align finset.sum_neg Finset.sum_neg
@[to_additive sum_pos']
theorem one_lt_prod' (h : β i β s, 1 β€ f i) (hs : β i β s, 1 < f i) : 1 < β i in s, f i :=
prod_const_one.symm.trans_lt <| prod_lt_prod' h hs
#align finset.one_lt_prod' Finset.one_lt_prod'
#align finset.sum_pos' Finset.sum_pos'
@[to_additive]
theorem prod_lt_one' (h : β i β s, f i β€ 1) (hs : β i β s, f i < 1) : β i in s, f i < 1 :=
prod_const_one.le.trans_lt' <| prod_lt_prod' h hs
#align finset.prod_lt_one' Finset.prod_lt_one'
#align finset.sum_neg' Finset.sum_neg'
@[to_additive]
theorem prod_eq_prod_iff_of_le {f g : ΞΉ β M} (h : β i β s, f i β€ g i) :
((β i in s, f i) = β i in s, g i) β β i β s, f i = g i := by
classical
revert h
refine Finset.induction_on s (fun _ β¦ β¨fun _ _ h β¦ False.elim (Finset.not_mem_empty _ h),
fun _ β¦ rflβ©) fun a s ha ih H β¦ ?_
specialize ih fun i β¦ H i β Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, β ih]
exact
mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a))
(Finset.prod_le_prod' fun i β¦ H i β Finset.mem_insert_of_mem)
#align finset.prod_eq_prod_iff_of_le Finset.prod_eq_prod_iff_of_le
#align finset.sum_eq_sum_iff_of_le Finset.sum_eq_sum_iff_of_le
variable [DecidableEq ΞΉ]
@[to_additive] lemma prod_sdiff_le_prod_sdiff :
β i in s \ t, f i β€ β i in t \ s, f i β β i in s, f i β€ β i in t, f i := by
rw [β mul_le_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
@[to_additive] lemma prod_sdiff_lt_prod_sdiff :
β i in s \ t, f i < β i in t \ s, f i β β i in s, f i < β i in t, f i := by
rw [β mul_lt_mul_iff_right, β prod_union (disjoint_sdiff_inter _ _), sdiff_union_inter,
β prod_union, inter_comm, sdiff_union_inter];
simpa only [inter_comm] using disjoint_sdiff_inter t s
end OrderedCancelCommMonoid
section LinearOrderedCancelCommMonoid
variable [LinearOrderedCancelCommMonoid M] {f g : ΞΉ β M} {s t : Finset ΞΉ}
@[to_additive exists_lt_of_sum_lt]
theorem exists_lt_of_prod_lt' (Hlt : β i in s, f i < β i in s, g i) : β i β s, f i < g i := by
contrapose! Hlt with Hle
exact prod_le_prod' Hle
#align finset.exists_lt_of_prod_lt' Finset.exists_lt_of_prod_lt'
#align finset.exists_lt_of_sum_lt Finset.exists_lt_of_sum_lt
@[to_additive exists_le_of_sum_le]
theorem exists_le_of_prod_le' (hs : s.Nonempty) (Hle : β i in s, f i β€ β i in s, g i) :
β i β s, f i β€ g i := by
contrapose! Hle with Hlt
exact prod_lt_prod_of_nonempty' hs Hlt
#align finset.exists_le_of_prod_le' Finset.exists_le_of_prod_le'
#align finset.exists_le_of_sum_le Finset.exists_le_of_sum_le
@[to_additive exists_pos_of_sum_zero_of_exists_nonzero]
theorem exists_one_lt_of_prod_one_of_exists_ne_one' (f : ΞΉ β M) (hβ : β i in s, f i = 1)
(hβ : β i β s, f i β 1) : β i β s, 1 < f i := by
contrapose! hβ
obtain β¨i, m, i_neβ© : β i β s, f i β 1 := hβ
apply ne_of_lt
calc
β j in s, f j < β j in s, 1 := prod_lt_prod' hβ β¨i, m, (hβ i m).lt_of_ne i_neβ©
_ = 1 := prod_const_one
#align finset.exists_one_lt_of_prod_one_of_exists_ne_one' Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
#align finset.exists_pos_of_sum_zero_of_exists_nonzero Finset.exists_pos_of_sum_zero_of_exists_nonzero
end LinearOrderedCancelCommMonoid
section OrderedCommSemiring
variable [OrderedCommSemiring R] {f g : ΞΉ β R} {s t : Finset ΞΉ}
open Classical
-- this is also true for an ordered commutative multiplicative monoid with zero
theorem prod_nonneg (h0 : β i β s, 0 β€ f i) : 0 β€ β i in s, f i :=
prod_induction f (fun i β¦ 0 β€ i) (fun _ _ ha hb β¦ mul_nonneg ha hb) zero_le_one h0
#align finset.prod_nonneg Finset.prod_nonneg
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`. See also `Finset.prod_le_prod'` for
the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
β i in s, f i β€ β i in s, g i := by
induction' s using Finset.induction with a s has ih h
Β· simp
Β· simp only [prod_insert has]
apply mul_le_mul
Β· exact h1 a (mem_insert_self a s)
Β· refine ih (fun x H β¦ h0 _ ?_) (fun x H β¦ h1 _ ?_) <;> exact mem_insert_of_mem H
Β· apply prod_nonneg fun x H β¦ h0 x (mem_insert_of_mem H)
Β· apply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))
#align finset.prod_le_prod Finset.prod_le_prod
/-- If all `f i`, `i β s`, are nonnegative and each `f i` is less than or equal to `g i`, then the
product of `f i` is less than or equal to the product of `g i`.
This is a variant (beta-reduced) version of the standard lemma `Finset.prod_le_prod`, convenient
for the `gcongr` tactic. -/
@[gcongr]
theorem _root_.GCongr.prod_le_prod (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ g i) :
s.prod f β€ s.prod g :=
s.prod_le_prod h0 h1
/-- If each `f i`, `i β s` belongs to `[0, 1]`, then their product is less than or equal to one.
See also `Finset.prod_le_one'` for the case of an ordered commutative multiplicative monoid. -/
theorem prod_le_one (h0 : β i β s, 0 β€ f i) (h1 : β i β s, f i β€ 1) : β i in s, f i β€ 1 := by
convert β prod_le_prod h0 h1
exact Finset.prod_const_one
#align finset.prod_le_one Finset.prod_le_one
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `OrderedCommSemiring`. -/
theorem prod_add_prod_le {i : ΞΉ} {f g h : ΞΉ β R} (hi : i β s) (h2i : g i + h i β€ f i)
(hgf : β j β s, j β i β g j β€ f j) (hhf : β j β s, j β i β h j β€ f j) (hg : β i β s, 0 β€ g i)
(hh : β i β s, 0 β€ h i) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)
Β· rw [right_distrib]
refine add_le_add ?_ ?_ <;>
Β· refine mul_le_mul_of_nonneg_left ?_ ?_
Β· refine prod_le_prod ?_ ?_
<;> simp (config := { contextual := true }) [*]
Β· try apply_assumption
try assumption
Β· apply prod_nonneg
simp only [and_imp, mem_sdiff, mem_singleton]
intro j h1j h2j
exact le_trans (hg j h1j) (hgf j h1j h2j)
#align finset.prod_add_prod_le Finset.prod_add_prod_le
end OrderedCommSemiring
section StrictOrderedCommSemiring
variable [StrictOrderedCommSemiring R] {f g : ΞΉ β R} {s : Finset ΞΉ}
-- This is also true for an ordered commutative multiplicative monoid with zero
theorem prod_pos (h0 : β i β s, 0 < f i) : 0 < β i in s, f i :=
prod_induction f (fun x β¦ 0 < x) (fun _ _ ha hb β¦ mul_pos ha hb) zero_lt_one h0
#align finset.prod_pos Finset.prod_pos
theorem prod_lt_prod (hf : β i β s, 0 < f i) (hfg : β i β s, f i β€ g i)
(hlt : β i β s, f i < g i) :
β i in s, f i < β i in s, g i := by
classical
obtain β¨i, hi, hiltβ© := hlt
rw [β insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]
apply mul_lt_mul hilt
Β· exact prod_le_prod (fun j hj => le_of_lt (hf j (mem_of_mem_erase hj)))
(fun _ hj β¦ hfg _ <| mem_of_mem_erase hj)
Β· exact prod_pos fun j hj => hf j (mem_of_mem_erase hj)
Β· exact le_of_lt <| (hf i hi).trans hilt
theorem prod_lt_prod_of_nonempty (hf : β i β s, 0 < f i) (hfg : β i β s, f i < g i)
(h_ne : s.Nonempty) :
β i in s, f i < β i in s, g i := by
apply prod_lt_prod hf fun i hi => le_of_lt (hfg i hi)
obtain β¨i, hiβ© := h_ne
exact β¨i, hi, hfg i hiβ©
end StrictOrderedCommSemiring
section CanonicallyOrderedCommSemiring
variable [CanonicallyOrderedCommSemiring R] {f g h : ΞΉ β R} {s : Finset ΞΉ} {i : ΞΉ}
/-- Note that the name is to match `CanonicallyOrderedCommSemiring.mul_pos`. -/
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
0 < β i in s, f i β (β i β s, (0 : R) < f i) :=
CanonicallyOrderedCommSemiring.multiset_prod_pos.trans Multiset.forall_mem_map_iff
#align canonically_ordered_comm_semiring.prod_pos CanonicallyOrderedCommSemiring.prod_pos
/-- If `g, h β€ f` and `g i + h i β€ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedCommSemiring`.
-/
theorem prod_add_prod_le' (hi : i β s) (h2i : g i + h i β€ f i) (hgf : β j β s, j β i β g j β€ f j)
(hhf : β j β s, j β i β h j β€ f j) : ((β i in s, g i) + β i in s, h i) β€ β i in s, f i := by
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi]
refine' le_trans _ (mul_le_mul_right' h2i _)
rw [right_distrib]
apply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>
simp only [and_imp, mem_sdiff, mem_singleton] <;>
intros <;>
apply_assumption <;>
assumption
#align finset.prod_add_prod_le' Finset.prod_add_prod_le'
end CanonicallyOrderedCommSemiring
end Finset
namespace Fintype
section OrderedCommMonoid
variable [Fintype ΞΉ] [OrderedCommMonoid M] {f : ΞΉ β M}
@[to_additive (attr := mono) sum_mono]
theorem prod_mono' : Monotone fun f : ΞΉ β M β¦ β i, f i := fun _ _ hfg β¦
Finset.prod_le_prod' fun x _ β¦ hfg x
#align fintype.prod_mono' Fintype.prod_mono'
#align fintype.sum_mono Fintype.sum_mono
@[to_additive sum_nonneg]
lemma one_le_prod (hf : 1 β€ f) : 1 β€ β i, f i := Finset.one_le_prod' Ξ» _ _ β¦ hf _
@[to_additive] lemma prod_le_one (hf : f β€ 1) : β i, f i β€ 1 := Finset.prod_le_one' Ξ» _ _ β¦ hf _
end OrderedCommMonoid
section OrderedCancelCommMonoid
variable [Fintype ΞΉ] [OrderedCancelCommMonoid M] {f : ΞΉ β M}
@[to_additive sum_strictMono]
theorem prod_strictMono' : StrictMono fun f : ΞΉ β M β¦ β x, f x :=
fun _ _ hfg β¦
let β¨hle, i, hltβ© := Pi.lt_def.mp hfg
Finset.prod_lt_prod' (fun i _ β¦ hle i) β¨i, Finset.mem_univ i, hltβ©
#align fintype.prod_strict_mono' Fintype.prod_strictMono'
#align fintype.sum_strict_mono Fintype.sum_strictMono
@[to_additive sum_pos]
lemma one_lt_prod (hf : 1 < f) : 1 < β i, f i :=
Finset.one_lt_prod' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive]
lemma prod_lt_one (hf : f < 1) : β i, f i < 1 :=
Finset.prod_lt_one' (Ξ» _ _ β¦ hf.le _) $ by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
lemma one_lt_prod_iff_of_one_le (hf : 1 β€ f) : 1 < β i, f i β 1 < f := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, one_lt_prod]
@[to_additive]
lemma prod_lt_one_iff_of_le_one (hf : f β€ 1) : β i, f i < 1 β f < 1 := by
obtain rfl | hf := hf.eq_or_lt <;> simp [*, prod_lt_one]
@[to_additive]
lemma prod_eq_one_iff_of_one_le (hf : 1 β€ f) : β i, f i = 1 β f = 1 := by
simpa only [(one_le_prod hf).not_gt_iff_eq, hf.not_gt_iff_eq]
using (one_lt_prod_iff_of_one_le hf).not
@[to_additive]
lemma prod_eq_one_iff_of_le_one (hf : f β€ 1) : β i, f i = 1 β f = 1 := by
simpa only [(prod_le_one hf).not_gt_iff_eq, hf.not_gt_iff_eq, eq_comm]
using (prod_lt_one_iff_of_le_one hf).not
end OrderedCancelCommMonoid
end Fintype
namespace WithTop
open Finset
/-- A product of finite numbers is still finite -/
theorem prod_lt_top [CommMonoidWithZero R] [NoZeroDivisors R] [Nontrivial R] [DecidableEq R] [LT R]
{s : Finset ΞΉ} {f : ΞΉ β WithTop R} (h : β i β s, f i β β€) : β i in s, f i < β€ :=
prod_induction f (fun a β¦ a < β€) (fun _ _ hβ hβ β¦ mul_lt_top' hβ hβ) (coe_lt_top 1)
fun a ha β¦ WithTop.lt_top_iff_ne_top.2 (h a ha)
#align with_top.prod_lt_top WithTop.prod_lt_top
/-- A sum of numbers is infinite iff one of them is infinite -/
theorem sum_eq_top_iff [AddCommMonoid M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i = β€ β β i β s, f i = β€ := by
induction s using Finset.cons_induction <;> simp [*]
#align with_top.sum_eq_top_iff WithTop.sum_eq_top_iff
/-- A sum of finite numbers is still finite -/
theorem sum_lt_top_iff [AddCommMonoid M] [LT M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i < β€ β β i β s, f i < β€ := by
| simp only [WithTop.lt_top_iff_ne_top, ne_eq, sum_eq_top_iff, not_exists, not_and] | /-- A sum of finite numbers is still finite -/
theorem sum_lt_top_iff [AddCommMonoid M] [LT M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i < β€ β β i β s, f i < β€ := by
| Mathlib.Algebra.BigOperators.Order.792_0.ewL52iF1Dz3xeLh | /-- A sum of finite numbers is still finite -/
theorem sum_lt_top_iff [AddCommMonoid M] [LT M] {s : Finset ΞΉ} {f : ΞΉ β WithTop M} :
β i in s, f i < β€ β β i β s, f i < β€ | Mathlib_Algebra_BigOperators_Order |
b x y : β
β’ logb b 0 = 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by | simp [logb] | @[simp]
theorem logb_zero : logb b 0 = 0 := by | Mathlib.Analysis.SpecialFunctions.Log.Base.48_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_zero : logb b 0 = 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
β’ logb b 1 = 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by | simp [logb] | @[simp]
theorem logb_one : logb b 1 = 0 := by | Mathlib.Analysis.SpecialFunctions.Log.Base.52_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_one : logb b 1 = 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
h : logb b b = 1
h' : log b = 0
β’ False | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by | simp [logb, h'] at h | lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by | Mathlib.Analysis.SpecialFunctions.Log.Base.60_0.egNyp4fdqSCAE7f | lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b xβ y x : β
β’ logb b |x| = logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by | rw [logb, logb, log_abs] | @[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by | Mathlib.Analysis.SpecialFunctions.Log.Base.63_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b xβ y x : β
β’ logb b (-x) = logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
| rw [β logb_abs x, β logb_abs (-x), abs_neg] | @[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.67_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
hx : x β 0
hy : y β 0
β’ logb b (x * y) = logb b x + logb b y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
| simp_rw [logb, log_mul hx hy, add_div] | theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.72_0.egNyp4fdqSCAE7f | theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
hx : x β 0
hy : y β 0
β’ logb b (x / y) = logb b x - logb b y | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
| simp_rw [logb, log_div hx hy, sub_div] | theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.76_0.egNyp4fdqSCAE7f | theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y | Mathlib_Analysis_SpecialFunctions_Log_Base |
b xβ y x : β
β’ logb b xβ»ΒΉ = -logb b x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by | simp [logb, neg_div] | @[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by | Mathlib.Analysis.SpecialFunctions.Log.Base.80_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
β’ (logb a b)β»ΒΉ = logb b a | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by | simp_rw [logb, inv_div] | theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by | Mathlib.Analysis.SpecialFunctions.Log.Base.84_0.egNyp4fdqSCAE7f | theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ (logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
| simp_rw [inv_logb] | theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.87_0.egNyp4fdqSCAE7f | theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ logb c (a * b) = logb c a + logb c b | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; | exact logb_mul hβ hβ | theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; | Mathlib.Analysis.SpecialFunctions.Log.Base.87_0.egNyp4fdqSCAE7f | theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ (logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
| simp_rw [inv_logb] | theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.92_0.egNyp4fdqSCAE7f | theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ logb c (a / b) = logb c a - logb c b | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; | exact logb_div hβ hβ | theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; | Mathlib.Analysis.SpecialFunctions.Log.Base.92_0.egNyp4fdqSCAE7f | theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by | rw [β inv_logb_mul_base hβ hβ c, inv_inv] | theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by | Mathlib.Analysis.SpecialFunctions.Log.Base.97_0.egNyp4fdqSCAE7f | theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b : β
hβ : a β 0
hβ : b β 0
c : β
β’ logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by | rw [β inv_logb_div_base hβ hβ c, inv_inv] | theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by | Mathlib.Analysis.SpecialFunctions.Log.Base.101_0.egNyp4fdqSCAE7f | theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b c : β
hβ : b β 0
hβ : b β 1
hβ : b β -1
β’ logb a b * logb b c = logb a c | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
| unfold logb | theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.105_0.egNyp4fdqSCAE7f | theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c | Mathlib_Analysis_SpecialFunctions_Log_Base |
bβ x y a b c : β
hβ : b β 0
hβ : b β 1
hβ : b β -1
β’ log b / log a * (log c / log b) = log c / log a | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
| rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)] | theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
| Mathlib.Analysis.SpecialFunctions.Log.Base.105_0.egNyp4fdqSCAE7f | theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
| have b_ne_zero : b β 0 | private theorem log_b_ne_zero : log b β 0 := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.120_0.egNyp4fdqSCAE7f | private theorem log_b_ne_zero : log b β 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
case b_ne_zero
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
β’ b β 0
b x y : β b_pos : 0 < b b_ne_one : b β 1 b_ne_zero : b β 0 β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; | linarith | private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; | Mathlib.Analysis.SpecialFunctions.Log.Base.120_0.egNyp4fdqSCAE7f | private theorem log_b_ne_zero : log b β 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
b_ne_zero : b β 0
β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
| have b_ne_minus_one : b β -1 | private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
| Mathlib.Analysis.SpecialFunctions.Log.Base.120_0.egNyp4fdqSCAE7f | private theorem log_b_ne_zero : log b β 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
case b_ne_minus_one
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
b_ne_zero : b β 0
β’ b β -1
b x y : β b_pos : 0 < b b_ne_one : b β 1 b_ne_zero : b β 0 b_ne_minus_one : b β -1 β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; | linarith | private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; | Mathlib.Analysis.SpecialFunctions.Log.Base.120_0.egNyp4fdqSCAE7f | private theorem log_b_ne_zero : log b β 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
b_ne_zero : b β 0
b_ne_minus_one : b β -1
β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
| simp [b_ne_one, b_ne_zero, b_ne_minus_one] | private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
| Mathlib.Analysis.SpecialFunctions.Log.Base.120_0.egNyp4fdqSCAE7f | private theorem log_b_ne_zero : log b β 0 | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
β’ logb b (b ^ x) = x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
| rw [logb, div_eq_iff, log_rpow b_pos] | @[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.125_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_rpow : logb b (b ^ x) = x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
β’ log b β 0 | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
| exact log_b_ne_zero b_pos b_ne_one | @[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
| Mathlib.Analysis.SpecialFunctions.Log.Base.125_0.egNyp4fdqSCAE7f | @[simp]
theorem logb_rpow : logb b (b ^ x) = x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ b ^ logb b x = |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
| apply log_injOn_pos | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ b ^ logb b x β Ioi 0
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ |x| β Ioi 0
case a b x y : β b_pos : 0 < b b_ne_one : b β 1 hx : x β 0 β’ log (b ^ logb b x) = log |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
| simp only [Set.mem_Ioi] | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ 0 < b ^ logb b x
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ |x| β Ioi 0
case a b x y : β b_pos : 0 < b b_ne_one : b β 1 hx : x β 0 β’ log (b ^ logb b x) = log |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
| apply rpow_pos_of_pos b_pos | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ |x| β Ioi 0
case a b x y : β b_pos : 0 < b b_ne_one : b β 1 hx : x β 0 β’ log (b ^ logb b x) = log |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
| simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff] | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ log (b ^ logb b x) = log |x| | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
| rw [log_rpow b_pos, logb, log_abs] | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
case a
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : x β 0
β’ log x / log b * log b = log x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
rw [log_rpow b_pos, logb, log_abs]
| field_simp [log_b_ne_zero b_pos b_ne_one] | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
rw [log_rpow b_pos, logb, log_abs]
| Mathlib.Analysis.SpecialFunctions.Log.Base.131_0.egNyp4fdqSCAE7f | theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : 0 < x
β’ b ^ logb b x = x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
rw [log_rpow b_pos, logb, log_abs]
field_simp [log_b_ne_zero b_pos b_ne_one]
#align real.rpow_logb_eq_abs Real.rpow_logb_eq_abs
@[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by
| rw [rpow_logb_eq_abs b_pos b_ne_one hx.ne'] | @[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by
| Mathlib.Analysis.SpecialFunctions.Log.Base.140_0.egNyp4fdqSCAE7f | @[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x | Mathlib_Analysis_SpecialFunctions_Log_Base |
b x y : β
b_pos : 0 < b
b_ne_one : b β 1
hx : 0 < x
β’ |x| = x | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
/-!
# Real logarithm base `b`
In this file we define `Real.logb` to be the logarithm of a real number in a given base `b`. We
define this as the division of the natural logarithms of the argument and the base, so that we have
a globally defined function with `logb b 0 = 0`, `logb b (-x) = logb b x` `logb 0 x = 0` and
`logb (-b) x = logb b x`.
We prove some basic properties of this function and its relation to `rpow`.
## Tags
logarithm, continuity
-/
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
/-- The real logarithm in a given base. As with the natural logarithm, we define `logb b x` to
be `logb b |x|` for `x < 0`, and `0` for `x = 0`.-/
-- @[pp_nodot] -- Porting note: removed
noncomputable def logb (b x : β) : β :=
log x / log b
#align real.logb Real.logb
theorem log_div_log : log x / log b = logb b x :=
rfl
#align real.log_div_log Real.log_div_log
@[simp]
theorem logb_zero : logb b 0 = 0 := by simp [logb]
#align real.logb_zero Real.logb_zero
@[simp]
theorem logb_one : logb b 1 = 0 := by simp [logb]
#align real.logb_one Real.logb_one
@[simp]
lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
div_self (log_pos hb).ne'
lemma logb_self_eq_one_iff : logb b b = 1 β b β 0 β§ b β 1 β§ b β -1 :=
Iff.trans β¨fun h h' => by simp [logb, h'] at h, div_selfβ© log_ne_zero
@[simp]
theorem logb_abs (x : β) : logb b |x| = logb b x := by rw [logb, logb, log_abs]
#align real.logb_abs Real.logb_abs
@[simp]
theorem logb_neg_eq_logb (x : β) : logb b (-x) = logb b x := by
rw [β logb_abs x, β logb_abs (-x), abs_neg]
#align real.logb_neg_eq_logb Real.logb_neg_eq_logb
theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by
simp_rw [logb, log_mul hx hy, add_div]
#align real.logb_mul Real.logb_mul
theorem logb_div (hx : x β 0) (hy : y β 0) : logb b (x / y) = logb b x - logb b y := by
simp_rw [logb, log_div hx hy, sub_div]
#align real.logb_div Real.logb_div
@[simp]
theorem logb_inv (x : β) : logb b xβ»ΒΉ = -logb b x := by simp [logb, neg_div]
#align real.logb_inv Real.logb_inv
theorem inv_logb (a b : β) : (logb a b)β»ΒΉ = logb b a := by simp_rw [logb, inv_div]
#align real.inv_logb Real.inv_logb
theorem inv_logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a * b) c)β»ΒΉ = (logb a c)β»ΒΉ + (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_mul hβ hβ
#align real.inv_logb_mul_base Real.inv_logb_mul_base
theorem inv_logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
(logb (a / b) c)β»ΒΉ = (logb a c)β»ΒΉ - (logb b c)β»ΒΉ := by
simp_rw [inv_logb]; exact logb_div hβ hβ
#align real.inv_logb_div_base Real.inv_logb_div_base
theorem logb_mul_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a * b) c = ((logb a c)β»ΒΉ + (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_mul_base hβ hβ c, inv_inv]
#align real.logb_mul_base Real.logb_mul_base
theorem logb_div_base {a b : β} (hβ : a β 0) (hβ : b β 0) (c : β) :
logb (a / b) c = ((logb a c)β»ΒΉ - (logb b c)β»ΒΉ)β»ΒΉ := by rw [β inv_logb_div_base hβ hβ c, inv_inv]
#align real.logb_div_base Real.logb_div_base
theorem mul_logb {a b c : β} (hβ : b β 0) (hβ : b β 1) (hβ : b β -1) :
logb a b * logb b c = logb a c := by
unfold logb
rw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr β¨hβ, hβ, hββ©)]
#align real.mul_logb Real.mul_logb
theorem div_logb {a b c : β} (hβ : c β 0) (hβ : c β 1) (hβ : c β -1) :
logb a c / logb b c = logb a b :=
div_div_div_cancel_left' _ _ <| log_ne_zero.mpr β¨hβ, hβ, hββ©
#align real.div_logb Real.div_logb
section BPosAndNeOne
variable (b_pos : 0 < b) (b_ne_one : b β 1)
private theorem log_b_ne_zero : log b β 0 := by
have b_ne_zero : b β 0; linarith
have b_ne_minus_one : b β -1; linarith
simp [b_ne_one, b_ne_zero, b_ne_minus_one]
@[simp]
theorem logb_rpow : logb b (b ^ x) = x := by
rw [logb, div_eq_iff, log_rpow b_pos]
exact log_b_ne_zero b_pos b_ne_one
#align real.logb_rpow Real.logb_rpow
theorem rpow_logb_eq_abs (hx : x β 0) : b ^ logb b x = |x| := by
apply log_injOn_pos
simp only [Set.mem_Ioi]
apply rpow_pos_of_pos b_pos
simp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]
rw [log_rpow b_pos, logb, log_abs]
field_simp [log_b_ne_zero b_pos b_ne_one]
#align real.rpow_logb_eq_abs Real.rpow_logb_eq_abs
@[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by
rw [rpow_logb_eq_abs b_pos b_ne_one hx.ne']
| exact abs_of_pos hx | @[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x := by
rw [rpow_logb_eq_abs b_pos b_ne_one hx.ne']
| Mathlib.Analysis.SpecialFunctions.Log.Base.140_0.egNyp4fdqSCAE7f | @[simp]
theorem rpow_logb (hx : 0 < x) : b ^ logb b x = x | Mathlib_Analysis_SpecialFunctions_Log_Base |
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