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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | all_goals clear s hs | 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) | case H0
⊢ RepbleByWord_aux ∅
case H1
⊢ ∀ {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:
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)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | . intros f hf
rw [Set.subset_empty_iff, support_empty_iff_id] at hf
subst hf
exact ⟨_, idHasWord⟩ | case H0
⊢ RepbleByWord_aux ∅
case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s) | case H1
⊢ ∀ {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:
case H0
⊢ RepbleByWord_aux ∅
case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | . intros a s _ _ hs_repble f hf
set g := (Equiv.swap a (f a)) * f with hg
suffices hgs : support g ⊆ s
obtain ⟨w, hw⟩ := hs_repble g hgs
obtain ⟨w_swap, hw_swap⟩ := swapHasWord a (f a)
use (w_swap ++ w)
simp [Word_mul, hw_swap, hw, hg]
. intro b hb
by_cases h : b = a
. subst b
absurd (unswap... | case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | clear s hs | 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) | case H1
⊢ ∀ {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:
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)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | intros f hf | case H0
⊢ RepbleByWord_aux ∅ | case H0
f : Equiv.Perm ℕ
hf : support f ⊆ ∅
⊢ RepbleByWord f | Please generate a tactic in lean4 to solve the state.
STATE:
case H0
⊢ RepbleByWord_aux ∅
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | rw [Set.subset_empty_iff, support_empty_iff_id] at hf | case H0
f : Equiv.Perm ℕ
hf : support f ⊆ ∅
⊢ RepbleByWord f | case H0
f : Equiv.Perm ℕ
hf : f = 1
⊢ RepbleByWord f | Please generate a tactic in lean4 to solve the state.
STATE:
case H0
f : Equiv.Perm ℕ
hf : support f ⊆ ∅
⊢ RepbleByWord f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | subst hf | case H0
f : Equiv.Perm ℕ
hf : f = 1
⊢ RepbleByWord f | case H0
⊢ RepbleByWord 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case H0
f : Equiv.Perm ℕ
hf : f = 1
⊢ RepbleByWord f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | exact ⟨_, idHasWord⟩ | case H0
⊢ RepbleByWord 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H0
⊢ RepbleByWord 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | intros a s _ _ hs_repble f hf | case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s) | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
⊢ RepbleByWord f | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
⊢ ∀ {a : ℕ} {s : Set ℕ}, a ∉ s → Set.Finite s → RepbleByWord_aux s → RepbleByWord_aux (insert a s)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | set g := (Equiv.swap a (f a)) * f with hg | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
⊢ RepbleByWord f | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ RepbleByWord f | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
⊢ RepbleByWord f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | suffices hgs : support g ⊆ s | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ RepbleByWord f | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
⊢ RepbleByWord f
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWo... | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ RepbleByWord f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | obtain ⟨w, hw⟩ := hs_repble g hgs | case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
⊢ RepbleByWord f
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWo... | case H1.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
⊢ RepbleByWord f
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝... | Please generate a tactic in lean4 to solve the state.
STATE:
case H1
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
⊢ RepbleByWord f
case hgs
a : ℕ
s... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | obtain ⟨w_swap, hw_swap⟩ := swapHasWord a (f a) | case H1.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
⊢ RepbleByWord f
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝... | case H1.intro.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.... | Please generate a tactic in lean4 to solve the state.
STATE:
case H1.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | use (w_swap ++ w) | case H1.intro.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.... | case h
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
... | Please generate a tactic in lean4 to solve the state.
STATE:
case H1.intro.intro
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toP... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | simp [Word_mul, hw_swap, hw, hg] | case h
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
... | case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ support g ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g ⊆ s
w : Word
hw : Word.toPerm w = g
w_s... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | . intro b hb
by_cases h : b = a
. subst b
absurd (unswap_support f a)
exact hb
exact Set.mem_of_mem_insert_of_ne (hf (unswap_support' _ _ hb)) h | case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ support g ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ support g ⊆ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | intro b hb | case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ support g ⊆ s | case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
⊢ support g ⊆ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | by_cases h : b = a | case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
⊢ b ∈ s | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : b = a
⊢ b ∈ s
case neg
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : Rep... | Please generate a tactic in lean4 to solve the state.
STATE:
case hgs
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
⊢ b ∈ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | . subst b
absurd (unswap_support f a)
exact hb | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : b = a
⊢ b ∈ s
case neg
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : Rep... | case neg
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : ¬b = a
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : b = a
⊢ b ∈ s
case neg
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | exact Set.mem_of_mem_insert_of_ne (hf (unswap_support' _ _ hb)) h | case neg
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : ¬b = a
⊢ b ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : ¬b = a
⊢ b ∈ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | subst b | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : b = a
⊢ b ∈ s | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
b : ℕ
hb : b ∈ support g
h : b = a
⊢ b ∈ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | absurd (unswap_support f a) | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ s | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ support (Equiv.swap a (f a) * f) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ s
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | exact hb | case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ support (Equiv.swap a (f a) * f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
a : ℕ
s : Set ℕ
a✝¹ : a ∉ s
a✝ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm ℕ
hf : support f ⊆ insert a s
g : Equiv.Perm ℕ := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a ∈ support g
⊢ a ∈ support (Equiv.swap a (f a) ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord | [153, 1] | [154, 48] | exact repbleByWord_aux _ hf f (subset_refl _) | f : Equiv.Perm ℕ
hf : Set.Finite (support f)
⊢ RepbleByWord f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
hf : Set.Finite (support f)
⊢ RepbleByWord f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions | [165, 1] | [166, 67] | rfl | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
p : α × α
⊢ p ∈ Inversions g ↔ (p.1 < p.2 ↔ g p.2 ≤ g p.1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
p : α × α
⊢ p ∈ Inversions g ↔ (p.1 < p.2 ↔ g p.2 ≤ g p.1)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions' | [168, 1] | [169, 56] | rfl | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ Inversions g ↔ (a < b ↔ g b ≤ g a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ Inversions g ↔ (a < b ↔ g b ≤ g a)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | not_mem_inversions_diag | [171, 1] | [172, 53] | simp [mem_inversions'] | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a : α
⊢ (a, a) ∉ Inversions g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a : α
⊢ (a, a) ∉ Inversions g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | simp only [mem_inversions'] | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ Inversions g ↔ (b, a) ∈ Inversions g | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ Inversions g ↔ (b, a) ∈ Inversions g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | wlog h : a < b | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | case inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | . cases' eq_or_lt_of_le (le_of_not_lt h) with h' h'
. simp [h']
. rw [this _ _ _ h'] | case inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
... | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | have h' : g a ≠ g b := fun hg => h.ne (g.injective hg) | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
h' : g a ≠ g b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | simp [h, h.not_lt, h'.symm.le_iff_lt] | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
h' : g a ≠ g b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a < b
h' : g a ≠ g b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | cases' eq_or_lt_of_le (le_of_not_lt h) with h' h' | case inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | case inr.inl
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b = a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
case inr.inr
α : Type u_1
inst✝ : LinearOrder... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | . simp [h'] | case inr.inl
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b = a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b)
case inr.inr
α : Type u_1
inst✝ : LinearOrder... | case inr.inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b < a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b = a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | . rw [this _ _ _ h'] | case inr.inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b < a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b < a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | simp [h'] | case inr.inl
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b = a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b = a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_inversions_symm | [174, 1] | [182, 40] | rw [this _ _ _ h'] | case inr.inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b < a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
this :
∀ {α : Type u_1} [inst : LinearOrder α] (g : Equiv.Perm α) (a b : α),
a < b → ((a < b ↔ g b ≤ g a) ↔ (b < a ↔ g a ≤ g b))
h : ¬a < b
h' : b < a
⊢ (a < b ↔ g b ≤ g a) ↔ (b < ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_support_of_inversion | [184, 1] | [189, 25] | simp only [mem_inversions', support, ne_eq, Set.mem_setOf_eq] at hab ⊢ | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : (a, b) ∈ Inversions g
⊢ a ∈ support g ∨ b ∈ support g | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
⊢ ¬g a = a ∨ ¬g b = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : (a, b) ∈ Inversions g
⊢ a ∈ support g ∨ b ∈ support g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_support_of_inversion | [184, 1] | [189, 25] | by_contra! hab' | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
⊢ ¬g a = a ∨ ¬g b = b | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
hab' : g a = a ∧ g b = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
⊢ ¬g a = a ∨ ¬g b = b
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_support_of_inversion | [184, 1] | [189, 25] | rw [hab'.1, hab'.2, ← not_le] at hab | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
hab' : g a = a ∧ g b = b
⊢ False | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : ¬b ≤ a ↔ b ≤ a
hab' : g a = a ∧ g b = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : a < b ↔ g b ≤ g a
hab' : g a = a ∧ g b = b
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_support_of_inversion | [184, 1] | [189, 25] | exact not_iff_self hab | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : ¬b ≤ a ↔ b ≤ a
hab' : g a = a ∧ g b = b
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
hab : ¬b ≤ a ↔ b ≤ a
hab' : g a = a ∧ g b = b
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | min_support_mem_inversions | [191, 1] | [195, 45] | rw [mem_inversions'] | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ (Nat.find hf, f⁻¹ (Nat.find hf)) ∈ Inversions f | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f⁻¹ (Nat.find hf) ↔ f (f⁻¹ (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⁻¹ (Nat.find hf)) ∈ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | min_support_mem_inversions | [191, 1] | [195, 45] | simp only [support_min_lt_inv_apply f hf, Equiv.Perm.apply_inv_self,
(support_min_lt_apply f hf).le] | f : Equiv.Perm ℕ
hf : ∃ n, f n ≠ n
⊢ Nat.find hf < f⁻¹ (Nat.find hf) ↔ f (f⁻¹ (Nat.find hf)) ≤ f (Nat.find hf) | no goals | 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) ↔ f (f⁻¹ (Nat.find hf)) ≤ f (Nat.find hf)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | suffices : ∃ m < n, n ≤ f m | f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, (m, n) ∈ Inversions f | f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
this : ∃ m < n, n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, (m, n) ∈ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | . obtain ⟨m, ⟨hmn, hm'⟩⟩ := this
refine' ⟨m, hmn, _⟩
simp [mem_inversions', hmn, (hn.trans_le hm').le] | f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
this : ∃ m < n, n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
this : ∃ m < n, n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | by_contra! h_lt | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
⊢ ∃ m < n, n ≤ f m
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | have key1 : Finset.image f (Finset.range (n + 1)) ⊆ Finset.range n | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ False | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | . intro _ hm
simp only [Finset.mem_image, Finset.mem_range] at hm ⊢
obtain ⟨m', ⟨hm', rfl⟩⟩ := hm
cases' Nat.lt_succ_iff_lt_or_eq.mp hm' with hm' hm'
. exact h_lt _ hm'
. simp [hm', hn] | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
⊢ False | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.r... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | replace key1 := Finset.card_le_card key1 | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
⊢ False | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.card (Finset.image (⇑f) (Finset.range (n + 1))) ≤ Finset.card (Finset.range n)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | simp [Finset.card_image_of_injective _ f.injective, Finset.card_range] at key1 | case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.card (Finset.image (⇑f) (Finset.range (n + 1))) ≤ Finset.card (Finset.range n)
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
key1 : Finset.card (Finset.image (⇑f) (Finset.range (n + 1))) ≤ Finset.card (Finset.range n)
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | obtain ⟨m, ⟨hmn, hm'⟩⟩ := this | f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
this : ∃ m < n, n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f | case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
this : ∃ m < n, n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | refine' ⟨m, hmn, _⟩ | case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f | case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ (m, n) ∈ Inversions f | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ ∃ m < n, (m, n) ∈ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | simp [mem_inversions', hmn, (hn.trans_le hm').le] | case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ (m, n) ∈ Inversions f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
m : ℕ
hmn : m < n
hm' : n ≤ f m
⊢ (m, n) ∈ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | intro _ hm | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : a✝ ∈ Finset.image (⇑f) (Finset.range (n + 1))
⊢ a✝ ∈ Finset.range n | Please generate a tactic in lean4 to solve the state.
STATE:
case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
⊢ Finset.image (⇑f) (Finset.range (n + 1)) ⊆ Finset.range n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | simp only [Finset.mem_image, Finset.mem_range] at hm ⊢ | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : a✝ ∈ Finset.image (⇑f) (Finset.range (n + 1))
⊢ a✝ ∈ Finset.range n | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : ∃ a < n + 1, f a = a✝
⊢ a✝ < n | Please generate a tactic in lean4 to solve the state.
STATE:
case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : a✝ ∈ Finset.image (⇑f) (Finset.range (n + 1))
⊢ a✝ ∈ Finset.range n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | obtain ⟨m', ⟨hm', rfl⟩⟩ := hm | case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : ∃ a < n + 1, f a = a✝
⊢ a✝ < n | case key1.intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm' : m' < n + 1
⊢ f m' < n | Please generate a tactic in lean4 to solve the state.
STATE:
case key1
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
a✝ : ℕ
hm : ∃ a < n + 1, f a = a✝
⊢ a✝ < n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | cases' Nat.lt_succ_iff_lt_or_eq.mp hm' with hm' hm' | case key1.intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm' : m' < n + 1
⊢ f m' < n | case key1.intro.intro.inl
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' < n
⊢ f m' < n
case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n | Please generate a tactic in lean4 to solve the state.
STATE:
case key1.intro.intro
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm' : m' < n + 1
⊢ f m' < n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | . exact h_lt _ hm' | case key1.intro.intro.inl
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' < n
⊢ f m' < n
case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n | case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n | Please generate a tactic in lean4 to solve the state.
STATE:
case key1.intro.intro.inl
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' < n
⊢ f m' < n
case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | . simp [hm', hn] | case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | exact h_lt _ hm' | case key1.intro.intro.inl
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' < n
⊢ f m' < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case key1.intro.intro.inl
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' < n
⊢ f m' < n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | simp [hm', hn] | case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case key1.intro.intro.inr
f : Equiv.Perm ℕ
n : ℕ
hn : f n < n
h_lt : ∀ m < n, f m < n
m' : ℕ
hm'✝ : m' < n + 1
hm' : m' = n
⊢ f m' < n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | rw [← not_iff_not] | f : Equiv.Perm ℕ
⊢ Inversions f = ∅ ↔ support f = ∅ | f : Equiv.Perm ℕ
⊢ ¬Inversions f = ∅ ↔ ¬support f = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
⊢ Inversions f = ∅ ↔ support f = ∅
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | push_neg | f : Equiv.Perm ℕ
⊢ ¬Inversions f = ∅ ↔ ¬support f = ∅ | f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) ↔ Set.Nonempty (support f) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
⊢ ¬Inversions f = ∅ ↔ ¬support f = ∅
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | constructor | f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) ↔ Set.Nonempty (support f) | case mp
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) → Set.Nonempty (support f)
case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f) | Please generate a tactic in lean4 to solve the state.
STATE:
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) ↔ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | . rintro ⟨⟨a, b⟩, hab⟩
cases' mem_support_of_inversion _ _ _ hab with ha hb
. exact ⟨_, ha⟩
. exact ⟨_, hb⟩ | case mp
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) → Set.Nonempty (support f)
case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f) | case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) → Set.Nonempty (support f)
case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | . exact fun hf => ⟨_, min_support_mem_inversions f hf⟩ | case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | rintro ⟨⟨a, b⟩, hab⟩ | case mp
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) → Set.Nonempty (support f) | case mp.intro.mk
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
⊢ Set.Nonempty (support f) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
f : Equiv.Perm ℕ
⊢ Set.Nonempty (Inversions f) → Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | cases' mem_support_of_inversion _ _ _ hab with ha hb | case mp.intro.mk
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
⊢ Set.Nonempty (support f) | case mp.intro.mk.inl
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
ha : a ∈ support f
⊢ Set.Nonempty (support f)
case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
⊢ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | . exact ⟨_, ha⟩ | case mp.intro.mk.inl
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
ha : a ∈ support f
⊢ Set.Nonempty (support f)
case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f) | case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.inl
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
ha : a ∈ support f
⊢ Set.Nonempty (support f)
case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | . exact ⟨_, hb⟩ | case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | exact ⟨_, ha⟩ | case mp.intro.mk.inl
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
ha : a ∈ support f
⊢ Set.Nonempty (support f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.inl
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
ha : a ∈ support f
⊢ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | exact ⟨_, hb⟩ | case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.inr
f : Equiv.Perm ℕ
a b : ℕ
hab : (a, b) ∈ Inversions f
hb : b ∈ support f
⊢ Set.Nonempty (support f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_empty_iff_support_empty | [214, 1] | [223, 57] | exact fun hf => ⟨_, min_support_mem_inversions f hf⟩ | case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
f : Equiv.Perm ℕ
⊢ Set.Nonempty (support f) → Set.Nonempty (Inversions f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | ext ⟨a, b⟩ | f g : Equiv.Perm ℕ
⊢ Inversions (f * g) = (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) ∆ Inversions g | case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔ (a, b) ∈ (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) ∆ Inversions g | Please generate a tactic in lean4 to solve the state.
STATE:
f g : Equiv.Perm ℕ
⊢ Inversions (f * g) = (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) ∆ Inversions g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | simp only [Set.mem_symmDiff, Set.mem_preimage, Equiv.prodCongr_apply, Prod.map] | case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔ (a, b) ∈ (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) ∆ Inversions g | case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔ (a, b) ∈ (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) ∆ Inversions g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | wlog h : a < b | case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | case h.mk.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk
f g : Equiv.Perm ℕ
a b : ℕ
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | . cases' (le_of_not_lt h).eq_or_lt with h' h'
. simp [h', not_mem_inversions_diag]
. repeat simp_rw [mem_inversions_symm _ a b, mem_inversions_symm _ (g a) (g b)]
exact this f g _ _ h' | case h.mk.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (... | f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
⊢ (a... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | cases' lt_or_le (g a) (g b) with hgab hgba | f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | case inl
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgab : g a < g b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Invers... | Please generate a tactic in lean4 to solve the state.
STATE:
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | . simp [mem_inversions', Set.mem_symmDiff, h, hgab, hgab.not_le] | case inl
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgab : g a < g b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Invers... | case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgab : g a < g b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | . simp [mem_inversions', Set.mem_symmDiff, h, hgba, hgba.not_lt] | case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | cases' (le_of_not_lt h).eq_or_lt with h' h' | case h.mk.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (... | case h.mk.inr.inl
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b = a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
⊢ (a... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | . simp [h', not_mem_inversions_diag] | case h.mk.inr.inl
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b = a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inl
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | . repeat simp_rw [mem_inversions_symm _ a b, mem_inversions_symm _ (g a) (g b)]
exact this f g _ _ h' | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | simp [h', not_mem_inversions_diag] | case h.mk.inr.inl
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b = a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inl
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | repeat simp_rw [mem_inversions_symm _ a b, mem_inversions_symm _ (g a) (g b)] | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (b, a) ∈ Inversions (f * g) ↔
(g b, g a) ∈ I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | exact this f g _ _ h' | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (b, a) ∈ Inversions (f * g) ↔
(g b, g a) ∈ I... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | simp_rw [mem_inversions_symm _ a b, mem_inversions_symm _ (g a) (g b)] | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ I... | case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
h' : b < a
⊢ (b, a) ∈ Inversions (f * g) ↔
(g b, g a) ∈ I... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mk.inr.inr
f g : Equiv.Perm ℕ
a b : ℕ
this :
∀ (f g : Equiv.Perm ℕ) (a b : ℕ),
a < b →
((a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f)
h : ¬a < b
... |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | simp [mem_inversions', Set.mem_symmDiff, h, hgab, hgab.not_le] | case inl
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgab : g a < g b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgab : g a < g b
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | inversions_comp | [230, 1] | [241, 67] | simp [mem_inversions', Set.mem_symmDiff, h, hgba, hgba.not_lt] | case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
f g : Equiv.Perm ℕ
a b : ℕ
h : a < b
hgba : g b ≤ g a
⊢ (a, b) ∈ Inversions (f * g) ↔
(g a, g b) ∈ Inversions f ∧ (a, b) ∉ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) ∉ Inversions f
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions | [249, 1] | [250, 70] | rfl | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
p : α × α
⊢ p ∈ StdInversions g ↔ p.1 < p.2 ∧ g p.2 ≤ g p.1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
p : α × α
⊢ p ∈ StdInversions g ↔ p.1 < p.2 ∧ g p.2 ≤ g p.1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions' | [252, 1] | [253, 59] | rfl | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ StdInversions g ↔ a < b ∧ g b ≤ g a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ StdInversions g ↔ a < b ∧ g b ≤ g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | rw [mem_stdinversions'] | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ StdInversions g ↔ a < b ∧ g b < g a | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ StdInversions g ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | by_cases h : a = b | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | . simp [h] | case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | . have h' : g a ≠ g b := fun hg => h (g.injective hg)
simp [h, h'.symm.le_iff_lt] | case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | simp [h] | case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | have h' : g a ≠ g b := fun hg => h (g.injective hg) | case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
h' : g a ≠ g b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | mem_stdinversions'' | [255, 1] | [261, 32] | simp [h, h'.symm.le_iff_lt] | case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
h' : g a ≠ g b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
h : ¬a = b
h' : g a ≠ g b
⊢ a < b ∧ g b ≤ g a ↔ a < b ∧ g b < g a
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | not_mem_stdinversions_diag | [263, 1] | [264, 59] | simp [mem_stdinversions'] | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a : α
⊢ (a, a) ∉ StdInversions g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a : α
⊢ (a, a) ∉ StdInversions g
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | stdinversions_inversions | [267, 1] | [278, 65] | ext ⟨a, b⟩ | α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
⊢ Inversions g = StdInversions g ∪ Prod.swap ⁻¹' StdInversions g | case h.mk
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
a b : α
⊢ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g ∪ Prod.swap ⁻¹' StdInversions g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝ : LinearOrder α
g : Equiv.Perm α
⊢ Inversions g = StdInversions g ∪ Prod.swap ⁻¹' StdInversions g
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.