url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
---|---|---|---|---|---|---|---|---|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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}
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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}
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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))
|
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
|
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))
|
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))
|
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
|
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))
|
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
|
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))
|
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))
|
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
|
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
|
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
|
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
|
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)
|
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)
|
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)
|
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)
|
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
|
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
|
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
|
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
|
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))
|
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)
|
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
|
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)
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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)
|
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)
|
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)
|
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)
|
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)
|
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)
|
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)
|
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)
|
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)
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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
|
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)
|
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