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stringclasses 147
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stringlengths 7
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stringlengths 1
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stringlengths 6
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https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_eq_empty | [34, 9] | [38, 12] | refine ⟨fun h ↦ ?_, fun h ↦ h.symm ▸ support'_one α⟩ | α : Type u_1
x y : α
f✝ f : Perm α
⊢ support' f = ∅ ↔ f = 1 | α : Type u_1
x y : α
f✝ f : Perm α
h : support' f = ∅
⊢ f = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
⊢ support' f = ∅ ↔ f = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_eq_empty | [34, 9] | [38, 12] | ext x | α : Type u_1
x y : α
f✝ f : Perm α
h : support' f = ∅
⊢ f = 1 | case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
h : support' f = ∅
x : α
⊢ f x = 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
h : support' f = ∅
⊢ f = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_eq_empty | [34, 9] | [38, 12] | simp only [support', ne_eq, eq_empty_iff_forall_not_mem, mem_setOf_eq, not_not] at h | case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
h : support' f = ∅
x : α
⊢ f x = 1 x | case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
h : ∀ (x : α), f x = x
⊢ f x = 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
h : support' f = ∅
x : α
⊢ f x = 1 x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_eq_empty | [34, 9] | [38, 12] | exact h x | case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
h : ∀ (x : α), f x = x
⊢ f x = 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
h : ∀ (x : α), f x = x
⊢ f x = 1 x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_ne_singleton | [40, 1] | [45, 50] | intro h | α : Type u_1
x y : α
f✝ f : Perm α
a : α
⊢ support' f ≠ {a} | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : support' f = {a}
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
a : α
⊢ support' f ≠ {a}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_ne_singleton | [40, 1] | [45, 50] | simp only [Set.ext_iff, mem_support', ne_eq, mem_singleton_iff] at h | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : support' f = {a}
⊢ False | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ∀ (x : α), ¬f x = x ↔ x = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : support' f = {a}
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_ne_singleton | [40, 1] | [45, 50] | specialize h (f.symm a) | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ∀ (x : α), ¬f x = x ↔ x = a
⊢ False | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f (f.symm a) = f.symm a ↔ f.symm a = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ∀ (x : α), ¬f x = x ↔ x = a
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_ne_singleton | [40, 1] | [45, 50] | rw [eq_comm] at h | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f (f.symm a) = f.symm a ↔ f.symm a = a
⊢ False | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f.symm a = f (f.symm a) ↔ f.symm a = a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f (f.symm a) = f.symm a ↔ f.symm a = a
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_ne_singleton | [40, 1] | [45, 50] | simp only [apply_symm_apply, not_iff_self] at h | α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f.symm a = f (f.symm a) ↔ f.symm a = a
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
a : α
h : ¬f.symm a = f (f.symm a) ↔ f.symm a = a
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | refine by_contra <| fun h ↦ support'_ne_singleton f a ?_ | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
⊢ ∃ b ∈ support' f, b ≠ a | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ support' f = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
⊢ ∃ b ∈ support' f, b ≠ a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | simp only [Set.ext_iff, mem_support', ne_eq, mem_singleton_iff] | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ support' f = {a} | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ ∀ (x : α), ¬f x = x ↔ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ support' f = {a}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | simp only [mem_support', ne_eq, not_exists, not_and, not_not] at h | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ ∀ (x : α), ¬f x = x ↔ x = a | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
⊢ ∀ (x : α), ¬f x = x ↔ x = a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ¬∃ b ∈ support' f, b ≠ a
⊢ ∀ (x : α), ¬f x = x ↔ x = a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | refine fun x ↦ ⟨fun h' ↦ h _ h', ?_⟩ | α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
⊢ ∀ (x : α), ¬f x = x ↔ x = a | α : Type u_1
x✝ y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
x : α
⊢ x = a → ¬f x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
⊢ ∀ (x : α), ¬f x = x ↔ x = a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | rintro rfl | α : Type u_1
x✝ y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
x : α
⊢ x = a → ¬f x = x | α : Type u_1
x✝ y : α
f : Perm α
x : α
ha : x ∈ support' f
h : ∀ (x_1 : α), ¬f x_1 = x_1 → x_1 = x
⊢ ¬f x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f : Perm α
a : α
ha : a ∈ support' f
h : ∀ (x : α), ¬f x = x → x = a
x : α
⊢ x = a → ¬f x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_ne_of_mem_support' | [47, 1] | [53, 11] | exact ha | α : Type u_1
x✝ y : α
f : Perm α
x : α
ha : x ∈ support' f
h : ∀ (x_1 : α), ¬f x_1 = x_1 → x_1 = x
⊢ ¬f x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f : Perm α
x : α
ha : x ∈ support' f
h : ∀ (x_1 : α), ¬f x_1 = x_1 → x_1 = x
⊢ ¬f x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_inv | [55, 9] | [57, 65] | ext x | α : Type u_1
x y : α
f✝ f : Perm α
⊢ support' f⁻¹ = support' f | case h
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
⊢ x ∈ support' f⁻¹ ↔ x ∈ support' f | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
⊢ support' f⁻¹ = support' f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_inv | [55, 9] | [57, 65] | rw [mem_support', ne_eq, mem_support', inv_eq_iff_eq, eq_comm] | case h
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
⊢ x ∈ support' f⁻¹ ↔ x ∈ support' f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x✝ y : α
f✝ f : Perm α
x : α
⊢ x ∈ support' f⁻¹ ↔ x ∈ support' f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_prod | [59, 1] | [63, 18] | refine fun x hx ↦ by_contra <| fun h ↦ hx ?_ | α : Type u_1
x y : α
f✝ f g : Perm α
⊢ support' (f * g) ⊆ support' f ∪ support' g | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : x ∉ support' f ∪ support' g
⊢ (f * g) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f g : Perm α
⊢ support' (f * g) ⊆ support' f ∪ support' g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_prod | [59, 1] | [63, 18] | simp only [mem_union, not_or, not_mem_support'] at h | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : x ∉ support' f ∪ support' g
⊢ (f * g) x = x | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ (f * g) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : x ∉ support' f ∪ support' g
⊢ (f * g) x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_prod | [59, 1] | [63, 18] | rw [coe_mul, Function.comp_apply] | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ (f * g) x = x | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ f (g x) = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ (f * g) x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_prod | [59, 1] | [63, 18] | simp [h.1, h.2] | α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ f (g x) = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ f g : Perm α
x : α
hx : x ∈ support' (f * g)
h : f x = x ∧ g x = x
⊢ f (g x) = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_swap | [69, 1] | [72, 65] | ext z | α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
⊢ support' (swap x y) = {x, y} | case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ z ∈ support' (swap x y) ↔ z ∈ {x, y} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
⊢ support' (swap x y) = {x, y}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_swap | [69, 1] | [72, 65] | simp only [support', ne_eq, mem_setOf_eq, mem_insert_iff, mem_singleton_iff] | case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ z ∈ support' (swap x y) ↔ z ∈ {x, y} | case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ ¬(swap x y) z = z ↔ z = x ∨ z = y | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ z ∈ support' (swap x y) ↔ z ∈ {x, y}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_swap | [69, 1] | [72, 65] | rw [← Ne.def, Equiv.swap_apply_ne_self_iff, and_iff_right hxy] | case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ ¬(swap x y) z = z ↔ z = x ∨ z = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f : Perm α
inst✝ : DecidableEq α
hxy : x ≠ y
z : α
⊢ ¬(swap x y) z = z ↔ z = x ∨ z = y
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_pair_of_isSwap | [102, 1] | [108, 29] | obtain ⟨x,y, hxy, rfl⟩ := hf | α : Type u_1
x y : α
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
hf : IsSwap f
⊢ ∃ p, f = swap p.lo p.hi | case intro.intro.intro
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
⊢ ∃ p, swap x y = swap p.lo p.hi | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
hf : IsSwap f
⊢ ∃ p, f = swap p.lo p.hi
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_pair_of_isSwap | [102, 1] | [108, 29] | obtain (h | h) := hxy.lt_or_lt | case intro.intro.intro
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
⊢ ∃ p, swap x y = swap p.lo p.hi | case intro.intro.intro.inl
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : x < y
⊢ ∃ p, swap x y = swap p.lo p.hi
case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap x y = swap p.lo p.hi | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
⊢ ∃ p, swap x y = swap p.lo p.hi
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_pair_of_isSwap | [102, 1] | [108, 29] | rw [Equiv.swap_comm] | case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap x y = swap p.lo p.hi | case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap y x = swap p.lo p.hi | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap x y = swap p.lo p.hi
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_pair_of_isSwap | [102, 1] | [108, 29] | exact ⟨Pair.mk y x h, rfl⟩ | case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap y x = swap p.lo p.hi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : y < x
⊢ ∃ p, swap y x = swap p.lo p.hi
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.exists_pair_of_isSwap | [102, 1] | [108, 29] | exact ⟨Pair.mk x y h, rfl⟩ | case intro.intro.intro.inl
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : x < y
⊢ ∃ p, swap x y = swap p.lo p.hi | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl
α : Type u_1
x✝ y✝ : α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x y : α
hxy : x ≠ y
h : x < y
⊢ ∃ p, swap x y = swap p.lo p.hi
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.swap_eq_swap | [112, 1] | [114, 8] | sorry | α : Type u_1
x✝ y✝ : α
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x x' y y' : α
⊢ swap x y = swap x' y' ↔ x = x' ∧ y = y' ∨ x = y' ∧ y = x' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y✝ : α
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : LinearOrder α
x x' y y' : α
⊢ swap x y = swap x' y' ↔ x = x' ∧ y = y' ∨ x = y' ∧ y = x'
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | WordForSwap_toPerm_succ | [21, 1] | [26, 6] | simp [WordForSwap, Word.toPerm] | i k : ℕ
⊢ Word.toPerm (WordForSwap i (k + 1)) =
Equiv.swap (i + (k + 1)) (i + (k + 2)) * Word.toPerm (WordForSwap i k) * Equiv.swap (i + (k + 1)) (i + (k + 2)) | i k : ℕ
⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) *
(List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + k + 1) (i + k + 1 + 1)) =
Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + (k + 1)) (i + (k + 2)) | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
⊢ Word.toPerm (WordForSwap i (k + 1)) =
Equiv.swap (i + (k + 1)) (i + (k + 2)) * Word.toPerm (WordForSwap i k) * Equiv.swap (i + (k + 1)) (i + (k + 2))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | WordForSwap_toPerm_succ | [21, 1] | [26, 6] | rfl | i k : ℕ
⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) *
(List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + k + 1) (i + k + 1 + 1)) =
Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + (k + 1)) (i + (k + 2)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
⊢ Equiv.swap (i + k + 1) (i + k + 1 + 1) *
(List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + k + 1) (i + k + 1 + 1)) =
Equiv.swap (i + (k + 1)) (i + (k + 2)) * List.prod (List.map (fun i => Equiv.swap i (i + 1)) (WordForSwap i k)) *
Equiv.swap (i + (k + 1)) (i + (k + 2))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | induction' k with k ih | i k : ℕ
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1)) | case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | . simp only [Nat.zero_eq, add_zero, Equiv.swap_self]
rfl | case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1)) | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | . rw [WordForSwap_toPerm_succ, ih]
change _ * _ * (Equiv.swap _ _)⁻¹ = _
rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left,
Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)] | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | simp only [Nat.zero_eq, add_zero, Equiv.swap_self] | case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1)) | case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i Nat.zero) = Equiv.swap i (i + (Nat.zero + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | rfl | case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
i : ℕ
⊢ Word.toPerm (WordForSwap i 0) = Equiv.swap i (i + (0 + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | rw [WordForSwap_toPerm_succ, ih] | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1)) | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) =
Equiv.swap i (i + (Nat.succ k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Word.toPerm (WordForSwap i (Nat.succ k)) = Equiv.swap i (i + (Nat.succ k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | change _ * _ * (Equiv.swap _ _)⁻¹ = _ | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) =
Equiv.swap i (i + (Nat.succ k + 1)) | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ =
Equiv.swap i (i + (Nat.succ k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * Equiv.swap (i + (k + 1)) (i + (k + 2)) =
Equiv.swap i (i + (Nat.succ k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | rw [← Equiv.swap_apply_apply, Equiv.swap_apply_left,
Equiv.swap_apply_of_ne_of_ne (by simp) (by simp)] | case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ =
Equiv.swap i (i + (Nat.succ k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ Equiv.swap (i + (k + 1)) (i + (k + 2)) * Equiv.swap i (i + (k + 1)) * (Equiv.swap (i + (k + 1)) (i + (k + 2)))⁻¹ =
Equiv.swap i (i + (Nat.succ k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | simp | i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 1)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | wordForSwap_eq_swap | [28, 1] | [36, 58] | simp | i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
i k : ℕ
ih : Word.toPerm (WordForSwap i k) = Equiv.swap i (i + (k + 1))
⊢ i ≠ i + (k + 2)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | Word_mul | [38, 1] | [39, 85] | simp only [Word.toPerm, FreeMonoid.ofList_append, map_mul, FreeMonoid.lift_ofList] | w v : Word
⊢ Word.toPerm (w ++ v) = Word.toPerm w * Word.toPerm v | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
w v : Word
⊢ Word.toPerm (w ++ v) = Word.toPerm w * Word.toPerm v
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | cases' le_or_lt i j with h h | i j : ℕ
⊢ RepbleByWord (Equiv.swap i j) | case inl
i j : ℕ
h : i ≤ j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | Please generate a tactic in lean4 to solve the state.
STATE:
i j : ℕ
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | cases' h.eq_or_lt with h h | case inl
i j : ℕ
h : i ≤ j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | case inl.inl
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ RepbleByWord (Equiv.swap i j)
case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
i j : ℕ
h : i ≤ j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | . use []
rw [idHasWord, h, Equiv.swap_self]
rfl | case inl.inl
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ RepbleByWord (Equiv.swap i j)
case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ RepbleByWord (Equiv.swap i j)
case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | . obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h
use WordForSwap i k
rw [wordForSwap_eq_swap, add_assoc] | case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | . obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h
use WordForSwap j k
rw [wordForSwap_eq_swap, add_assoc, Equiv.swap_comm] | case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | use [] | case inl.inl
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ RepbleByWord (Equiv.swap i j) | case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ Word.toPerm [] = Equiv.swap i j | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | rw [idHasWord, h, Equiv.swap_self] | case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ Word.toPerm [] = Equiv.swap i j | case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ 1 = Equiv.refl ℕ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ Word.toPerm [] = Equiv.swap i j
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | rfl | case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ 1 = Equiv.refl ℕ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
i j : ℕ
h✝ : i ≤ j
h : i = j
⊢ 1 = Equiv.refl ℕ
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h | case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j) | case inl.inr.intro
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ RepbleByWord (Equiv.swap i (i + k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
i j : ℕ
h✝ : i ≤ j
h : i < j
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | use WordForSwap i k | case inl.inr.intro
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ RepbleByWord (Equiv.swap i (i + k + 1)) | case h
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.intro
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ RepbleByWord (Equiv.swap i (i + k + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | rw [wordForSwap_eq_swap, add_assoc] | case h
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
i k : ℕ
h✝ : i ≤ i + k + 1
h : i < i + k + 1
⊢ Word.toPerm (WordForSwap i k) = Equiv.swap i (i + k + 1)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_lt h | case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j) | case inr.intro
j k : ℕ
h : j < j + k + 1
⊢ RepbleByWord (Equiv.swap (j + k + 1) j) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
i j : ℕ
h : j < i
⊢ RepbleByWord (Equiv.swap i j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | use WordForSwap j k | case inr.intro
j k : ℕ
h : j < j + k + 1
⊢ RepbleByWord (Equiv.swap (j + k + 1) j) | case h
j k : ℕ
h : j < j + k + 1
⊢ Word.toPerm (WordForSwap j k) = Equiv.swap (j + k + 1) j | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
j k : ℕ
h : j < j + k + 1
⊢ RepbleByWord (Equiv.swap (j + k + 1) j)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | swapHasWord | [49, 1] | [60, 57] | rw [wordForSwap_eq_swap, add_assoc, Equiv.swap_comm] | case h
j k : ℕ
h : j < j + k + 1
⊢ Word.toPerm (WordForSwap j k) = Equiv.swap (j + k + 1) j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
j k : ℕ
h : j < j + k + 1
⊢ Word.toPerm (WordForSwap j k) = Equiv.swap (j + k + 1) j
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_inv | [68, 1] | [72, 43] | ext a | α : Type u_1
f : Equiv.Perm α
⊢ support f = support f⁻¹ | case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ a ∈ support f ↔ a ∈ support f⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : Equiv.Perm α
⊢ support f = support f⁻¹
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_inv | [68, 1] | [72, 43] | simp [support, not_iff_not] | case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ a ∈ support f ↔ a ∈ support f⁻¹ | case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ f a = a ↔ f⁻¹ a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ a ∈ support f ↔ a ∈ support f⁻¹
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_inv | [68, 1] | [72, 43] | rw [← Equiv.Perm.eq_inv_iff_eq, eq_comm] | case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ f a = a ↔ f⁻¹ a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
f : Equiv.Perm α
a : α
⊢ f a = a ↔ f⁻¹ a = a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | set n := Nat.find hf with hn | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f⁻¹ (Nat.find hf) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
⊢ n < f⁻¹ n | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f⁻¹ (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | obtain ⟨hn', hn''⟩ := (Nat.find_eq_iff hf).mp hn.symm | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
⊢ n < f⁻¹ n | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : ∀ n_1 < n, ¬f n_1 ≠ n_1
⊢ n < f⁻¹ n | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
⊢ n < f⁻¹ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | specialize hn'' (f⁻¹ n) | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : ∀ n_1 < n, ¬f n_1 ≠ n_1
⊢ n < f⁻¹ n | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : f⁻¹ n < n → ¬f (f⁻¹ n) ≠ f⁻¹ n
⊢ n < f⁻¹ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : ∀ n_1 < n, ¬f n_1 ≠ n_1
⊢ n < f⁻¹ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | rw [imp_not_comm, not_lt, f.apply_inv_self] at hn'' | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : f⁻¹ n < n → ¬f (f⁻¹ n) ≠ f⁻¹ n
⊢ n < f⁻¹ n | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : f⁻¹ n < n → ¬f (f⁻¹ n) ≠ f⁻¹ n
⊢ n < f⁻¹ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | rw [ne_eq, ← Equiv.Perm.eq_inv_iff_eq] at hn' | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : ¬n = f⁻¹ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : f n ≠ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_inv_apply | [74, 1] | [81, 38] | apply lt_of_le_of_ne (hn'' hn') hn' | case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : ¬n = f⁻¹ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
n : ℕ := Nat.find hf
hn : n = Nat.find hf
hn' : ¬n = f⁻¹ n
hn'' : n ≠ f⁻¹ n → n ≤ f⁻¹ n
⊢ n < f⁻¹ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | have hg : ∃ n, f⁻¹ n ≠ n | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | . simp_rw [ne_comm, ne_eq, Equiv.Perm.eq_inv_iff_eq]
exact hf | case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | have hf_finv : ∀ n, f n ≠ n ↔ f⁻¹ n ≠ n | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | . simp_rw [ne_comm, ne_eq, Equiv.Perm.eq_inv_iff_eq, eq_comm]
exact fun n ↦ trivial | case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | have key : Nat.find hf = Nat.find hg | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf) | case key
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf = Nat.find hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | . simp_rw [Nat.find_eq_iff, hf_finv, ← Nat.find_eq_iff hg] | case key
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf = Nat.find hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf) | Please generate a tactic in lean4 to solve the state.
STATE:
case key
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf = Nat.find hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | rw [key] | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hg < f (Nat.find hg) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hf < f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | have := support_min_lt_inv_apply f⁻¹ hg | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hg < f (Nat.find hg) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f⁻¹⁻¹ (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
⊢ Nat.find hg < f (Nat.find hg)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | rw [inv_inv] at this | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f⁻¹⁻¹ (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg) | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f⁻¹⁻¹ (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | exact this | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
key : Nat.find hf = Nat.find hg
this : Nat.find hg < f (Nat.find hg)
⊢ Nat.find hg < f (Nat.find hg)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | simp_rw [ne_comm, ne_eq, Equiv.Perm.eq_inv_iff_eq] | case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, f⁻¹ n ≠ n | case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, ¬f n = n | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, f⁻¹ n ≠ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | exact hf | case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, ¬f n = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ ∃ n, ¬f n = n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | simp_rw [ne_comm, ne_eq, Equiv.Perm.eq_inv_iff_eq, eq_comm] | case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n | case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ℕ → True | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | exact fun n ↦ trivial | case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ℕ → True | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf_finv
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
⊢ ℕ → True
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_min_lt_apply | [83, 1] | [96, 13] | simp_rw [Nat.find_eq_iff, hf_finv, ← Nat.find_eq_iff hg] | case key
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf = Nat.find hg | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case key
f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
hg : ∃ n, f⁻¹ n ≠ n
hf_finv : ∀ (n : ℕ), f n ≠ n ↔ f⁻¹ n ≠ n
⊢ Nat.find hf = Nat.find hg
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_empty_iff_id | [98, 1] | [102, 31] | simp only [Set.eq_empty_iff_forall_not_mem, support,
ne_eq, Set.mem_setOf_eq, not_not] | α : Type u_1
f : Equiv.Perm α
⊢ support f = ∅ ↔ f = 1 | α : Type u_1
f : Equiv.Perm α
⊢ (∀ (x : α), f x = x) ↔ f = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : Equiv.Perm α
⊢ support f = ∅ ↔ f = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | support_empty_iff_id | [98, 1] | [102, 31] | exact Iff.symm Equiv.ext_iff | α : Type u_1
f : Equiv.Perm α
⊢ (∀ (x : α), f x = x) ↔ f = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
f : Equiv.Perm α
⊢ (∀ (x : α), f x = x) ↔ f = 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support | [104, 1] | [106, 22] | simp [support] at * | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ a ∉ support (Equiv.swap a (f a) * f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ a ∉ support (Equiv.swap a (f a) * f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | simp [support] at * | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ support (Equiv.swap a (f a) * f) ⊆ support f | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ ∀ (a_1 : α), ¬(Equiv.swap a (f a)) (f a_1) = a_1 → ¬f a_1 = a_1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ support (Equiv.swap a (f a) * f) ⊆ support f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | intro b | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ ∀ (a_1 : α), ¬(Equiv.swap a (f a)) (f a_1) = a_1 → ¬f a_1 = a_1 | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ ¬(Equiv.swap a (f a)) (f b) = b → ¬f b = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
⊢ ∀ (a_1 : α), ¬(Equiv.swap a (f a)) (f a_1) = a_1 → ¬f a_1 = a_1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | contrapose! | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ ¬(Equiv.swap a (f a)) (f b) = b → ¬f b = b | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ f b = b → (Equiv.swap a (f a)) (f b) = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ ¬(Equiv.swap a (f a)) (f b) = b → ¬f b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | intro hbf | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ f b = b → (Equiv.swap a (f a)) (f b) = b | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) (f b) = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
⊢ f b = b → (Equiv.swap a (f a)) (f b) = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | rw [hbf] | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) (f b) = b | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) b = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) (f b) = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | by_cases h : b = a | α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) b = b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : b = a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | . subst b
simp [hbf] | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : b = a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b | case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : b = a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | by_cases h' : b = f a | case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : b = f a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | . subst b
simp
apply f.injective hbf.symm | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : b = f a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b | case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : b = f a
⊢ (Equiv.swap a (f a)) b = b
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | apply Equiv.swap_apply_of_ne_of_ne h h' | case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : ¬b = f a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | subst b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : b = a
⊢ (Equiv.swap a (f a)) b = b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f a = a
⊢ (Equiv.swap a (f a)) a = a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : b = a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | simp [hbf] | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f a = a
⊢ (Equiv.swap a (f a)) a = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f a = a
⊢ (Equiv.swap a (f a)) a = a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | subst b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : b = f a
⊢ (Equiv.swap a (f a)) b = b | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ (Equiv.swap a (f a)) (f a) = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a b : α
hbf : f b = b
h : ¬b = a
h' : b = f a
⊢ (Equiv.swap a (f a)) b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | simp | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ (Equiv.swap a (f a)) (f a) = f a | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ a = f a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ (Equiv.swap a (f a)) (f a) = f a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | unswap_support' | [108, 1] | [122, 42] | apply f.injective hbf.symm | case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ a = f a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : DecidableEq α
f : Equiv.Perm α
a : α
hbf : f (f a) = f a
h : ¬f a = a
⊢ a = f a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | apply @Set.Finite.induction_on _ RepbleByWord_aux _ hs | s : Set ℕ
hs : Set.Finite s
⊢ RepbleByWord_aux s | case H0
s : Set ℕ
hs : Set.Finite s
⊢ RepbleByWord_aux ∅
case H1
s : Set ℕ
hs : Set.Finite s
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s) | Please generate a tactic in lean4 to solve the state.
STATE:
s : Set ℕ
hs : Set.Finite s
⊢ RepbleByWord_aux s
TACTIC:
|
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