Datasets:
internlm
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Lean-Github

Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
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stringlengths
6
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6
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 β„• hs : Set.Finite s ⊒ RepbleByWord_aux βˆ… case H1 s : Set β„• hs : Set.Finite s ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
case H0 ⊒ RepbleByWord_aux βˆ… case H1 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
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 βˆ… case H1 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
case H1 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
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 βŠ† 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_support f a) exact hb exact Set.mem_of_mem_insert_of_ne (hf (unswap_support' _ _ hb)) h
case H1 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
clear s hs
case H1 s : Set β„• hs : Set.Finite s ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
case H1 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
intros f hf
case H0 ⊒ RepbleByWord_aux βˆ…
case H0 f : Equiv.Perm β„• hf : support f βŠ† βˆ… ⊒ RepbleByWord f
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 β„• hf : support f βŠ† βˆ… ⊒ RepbleByWord f
case H0 f : Equiv.Perm β„• hf : f = 1 ⊒ RepbleByWord f
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
subst hf
case H0 f : Equiv.Perm β„• hf : f = 1 ⊒ RepbleByWord f
case H0 ⊒ RepbleByWord 1
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
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 ⊒ βˆ€ {a : β„•} {s : Set β„•}, a βˆ‰ s β†’ Set.Finite s β†’ RepbleByWord_aux s β†’ RepbleByWord_aux (insert a s)
case H1 a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s ⊒ RepbleByWord f
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s ⊒ RepbleByWord f
case H1 a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ RepbleByWord f
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
suffices hgs : support g βŠ† s
case H1 a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ RepbleByWord f
case H1 a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
case H1.intro a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† 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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
case H1.intro.intro a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g w_swap : Word hw_swap : Word.toPerm w_swap = Equiv.swap a (f a) ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g w_swap : Word hw_swap : Word.toPerm w_swap = Equiv.swap a (f a) ⊒ RepbleByWord f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
case h a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g w_swap : Word hw_swap : Word.toPerm w_swap = Equiv.swap a (f a) ⊒ Word.toPerm (w_swap ++ w) = f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hgs : support g βŠ† s w : Word hw : Word.toPerm w = g w_swap : Word hw_swap : Word.toPerm w_swap = Equiv.swap a (f a) ⊒ Word.toPerm (w_swap ++ w) = f case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† 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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
intro b hb
case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f ⊒ support g βŠ† s
case hgs a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g ⊒ b ∈ s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g ⊒ b ∈ s
case pos a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : b = a ⊒ b ∈ s case neg a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : Β¬b = a ⊒ b ∈ s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : b = a ⊒ b ∈ s case neg a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : Β¬b = a ⊒ b ∈ s
case neg a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : Β¬b = a ⊒ b ∈ s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : Β¬b = a ⊒ b ∈ s
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
subst b
case pos a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f b : β„• hb : b ∈ support g h : b = a ⊒ b ∈ s
case pos a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hb : a ∈ support g ⊒ a ∈ s
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 : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := Equiv.swap a (f a) * f hg : g = Equiv.swap a (f a) * f hb : a ∈ support g ⊒ a ∈ s
case pos a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := 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)
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
repbleByWord_aux
[130, 1]
[150, 7]
exact hb
case pos a : β„• s : Set β„• a✝¹ : a βˆ‰ s a✝ : Set.Finite s hs_repble : RepbleByWord_aux s f : Equiv.Perm β„• hf : support f βŠ† insert a s g : Equiv.Perm β„• := 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
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 β„• hf : Set.Finite (support f) ⊒ RepbleByWord f
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_inversions
[165, 1]
[166, 67]
rfl
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± p : Ξ± Γ— Ξ± ⊒ p ∈ Inversions g ↔ (p.1 < p.2 ↔ g p.2 ≀ g p.1)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_inversions'
[168, 1]
[169, 56]
rfl
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a, b) ∈ Inversions g ↔ (a < b ↔ g b ≀ g a)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
not_mem_inversions_diag
[171, 1]
[172, 53]
simp [mem_inversions']
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a : Ξ± ⊒ (a, a) βˆ‰ Inversions g
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_inversions_symm
[174, 1]
[182, 40]
simp only [mem_inversions']
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a, b) ∈ Inversions g ↔ (b, a) ∈ Inversions g
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (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]
wlog h : a < b
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
case inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b) Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± 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]
. cases' eq_or_lt_of_le (le_of_not_lt h) with h' h' . simp [h'] . rw [this _ _ _ h']
case inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b) Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a < b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± 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 β‰  g b := fun hg => h.ne (g.injective hg)
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a < b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a < b h' : g a β‰  g 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, h.not_lt, h'.symm.le_iff_lt]
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a < b h' : g a β‰  g b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
no goals
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
case inr.inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b = a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b) case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b < a ⊒ (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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b = a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b) case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b < a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b < a ⊒ (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]
. rw [this _ _ _ h']
case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b < a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_inversions_symm
[174, 1]
[182, 40]
simp [h']
case inr.inl Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b = a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_inversions_symm
[174, 1]
[182, 40]
rw [this _ _ _ h']
case inr.inr Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± this : βˆ€ {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±), a < b β†’ ((a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)) h : Β¬a < b h' : b < a ⊒ (a < b ↔ g b ≀ g a) ↔ (b < a ↔ g a ≀ g b)
no goals
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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : a < b ↔ g b ≀ g a ⊒ Β¬g a = a ∨ Β¬g b = b
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_support_of_inversion
[184, 1]
[189, 25]
by_contra! hab'
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : a < b ↔ g b ≀ g a ⊒ Β¬g a = a ∨ Β¬g b = b
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : a < b ↔ g b ≀ g a hab' : g a = a ∧ g b = b ⊒ False
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
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : a < b ↔ g b ≀ g a hab' : g a = a ∧ g b = b ⊒ False
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : Β¬b ≀ a ↔ b ≀ a hab' : g a = a ∧ g b = b ⊒ False
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_support_of_inversion
[184, 1]
[189, 25]
exact not_iff_self hab
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± hab : Β¬b ≀ a ↔ b ≀ a hab' : g a = a ∧ g b = b ⊒ False
no goals
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 β„• hf : βˆƒ n, f n β‰  n ⊒ (Nat.find hf, f⁻¹ (Nat.find hf)) ∈ Inversions f
f : Equiv.Perm β„• hf : βˆƒ n, f n β‰  n ⊒ Nat.find hf < f⁻¹ (Nat.find hf) ↔ f (f⁻¹ (Nat.find hf)) ≀ f (Nat.find hf)
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 β„• hf : βˆƒ n, f n β‰  n ⊒ Nat.find hf < f⁻¹ (Nat.find hf) ↔ f (f⁻¹ (Nat.find hf)) ≀ f (Nat.find hf)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
exists_inversion_of_apply_lt
[197, 1]
[212, 81]
suffices : βˆƒ m < n, n ≀ f m
f : Equiv.Perm β„• n : β„• hn : f n < n ⊒ βˆƒ m < n, (m, n) ∈ Inversions f
f : Equiv.Perm β„• n : β„• hn : f n < n this : βˆƒ m < n, n ≀ f m ⊒ βˆƒ m < n, (m, n) ∈ Inversions f case this f : Equiv.Perm β„• n : β„• hn : f n < n ⊒ βˆƒ m < n, n ≀ f m
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 β„• n : β„• hn : f n < n this : βˆƒ m < n, n ≀ f m ⊒ βˆƒ m < n, (m, n) ∈ Inversions f case this f : Equiv.Perm β„• n : β„• hn : f n < n ⊒ βˆƒ m < n, n ≀ f m
case this f : Equiv.Perm β„• n : β„• hn : f n < n ⊒ βˆƒ m < n, n ≀ f m
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 β„• n : β„• hn : f n < n ⊒ βˆƒ m < n, n ≀ f m
case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n ⊒ False
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)) βŠ† Finset.range n
case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n ⊒ False
case key1 f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n ⊒ Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n ⊒ False
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n ⊒ Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n ⊒ False
case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n ⊒ False
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n ⊒ False
case this f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.card (Finset.image (⇑f) (Finset.range (n + 1))) ≀ Finset.card (Finset.range n) ⊒ False
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n key1 : Finset.card (Finset.image (⇑f) (Finset.range (n + 1))) ≀ Finset.card (Finset.range n) ⊒ False
no goals
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 β„• n : β„• hn : f n < n this : βˆƒ m < n, n ≀ f m ⊒ βˆƒ m < n, (m, n) ∈ Inversions f
case intro.intro f : Equiv.Perm β„• n : β„• hn : f n < n m : β„• hmn : m < n hm' : n ≀ f m ⊒ βˆƒ m < n, (m, n) ∈ Inversions f
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 β„• n : β„• hn : f n < n m : β„• hmn : m < n hm' : n ≀ f m ⊒ βˆƒ m < n, (m, n) ∈ Inversions f
case intro.intro f : Equiv.Perm β„• n : β„• hn : f n < n m : β„• hmn : m < n hm' : n ≀ f m ⊒ (m, n) ∈ Inversions f
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 β„• n : β„• hn : f n < n m : β„• hmn : m < n hm' : n ≀ f m ⊒ (m, n) ∈ Inversions f
no goals
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n ⊒ Finset.image (⇑f) (Finset.range (n + 1)) βŠ† Finset.range n
case key1 f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n a✝ : β„• hm : a✝ ∈ Finset.image (⇑f) (Finset.range (n + 1)) ⊒ a✝ ∈ Finset.range n
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n a✝ : β„• hm : a✝ ∈ Finset.image (⇑f) (Finset.range (n + 1)) ⊒ a✝ ∈ Finset.range n
case key1 f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n a✝ : β„• hm : βˆƒ a < n + 1, f a = a✝ ⊒ a✝ < n
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n a✝ : β„• hm : βˆƒ a < n + 1, f a = a✝ ⊒ a✝ < n
case key1.intro.intro f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm' : m' < n + 1 ⊒ f m' < n
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm' : m' < n + 1 ⊒ f m' < n
case key1.intro.intro.inl f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' < n ⊒ f m' < n case key1.intro.intro.inr f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' = n ⊒ f m' < n
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' < n ⊒ f m' < n case key1.intro.intro.inr f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' = n ⊒ f m' < n
case key1.intro.intro.inr f : Equiv.Perm β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' = n ⊒ f 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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' = n ⊒ f m' < n
no goals
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' < n ⊒ f m' < n
no goals
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 β„• n : β„• hn : f n < n h_lt : βˆ€ m < n, f m < n m' : β„• hm'✝ : m' < n + 1 hm' : m' = n ⊒ f m' < n
no goals
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 β„• ⊒ Inversions f = βˆ… ↔ support f = βˆ…
f : Equiv.Perm β„• ⊒ Β¬Inversions f = βˆ… ↔ Β¬support f = βˆ…
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 β„• ⊒ Β¬Inversions f = βˆ… ↔ Β¬support f = βˆ…
f : Equiv.Perm β„• ⊒ Set.Nonempty (Inversions f) ↔ Set.Nonempty (support f)
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
inversions_empty_iff_support_empty
[214, 1]
[223, 57]
constructor
f : Equiv.Perm β„• ⊒ Set.Nonempty (Inversions f) ↔ Set.Nonempty (support f)
case mp f : Equiv.Perm β„• ⊒ Set.Nonempty (Inversions f) β†’ Set.Nonempty (support f) case mpr f : Equiv.Perm β„• ⊒ Set.Nonempty (support f) β†’ Set.Nonempty (Inversions f)
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 β„• ⊒ Set.Nonempty (Inversions f) β†’ Set.Nonempty (support f) case mpr f : Equiv.Perm β„• ⊒ Set.Nonempty (support f) β†’ Set.Nonempty (Inversions f)
case mpr f : Equiv.Perm β„• ⊒ Set.Nonempty (support f) β†’ Set.Nonempty (Inversions f)
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 β„• ⊒ Set.Nonempty (support f) β†’ Set.Nonempty (Inversions f)
no goals
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 β„• ⊒ Set.Nonempty (Inversions f) β†’ Set.Nonempty (support f)
case mp.intro.mk f : Equiv.Perm β„• a b : β„• hab : (a, b) ∈ Inversions f ⊒ Set.Nonempty (support f)
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 β„• a b : β„• hab : (a, b) ∈ Inversions f ⊒ Set.Nonempty (support f)
case mp.intro.mk.inl f : Equiv.Perm β„• a b : β„• hab : (a, b) ∈ Inversions f ha : a ∈ support f ⊒ Set.Nonempty (support f) case mp.intro.mk.inr f : Equiv.Perm β„• a b : β„• hab : (a, b) ∈ Inversions f hb : b ∈ support f ⊒ Set.Nonempty (support f)
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 β„• a b : β„• hab : (a, b) ∈ Inversions f ha : a ∈ support f ⊒ Set.Nonempty (support f) case mp.intro.mk.inr f : Equiv.Perm β„• a b : β„• hab : (a, b) ∈ Inversions f hb : b ∈ support f ⊒ Set.Nonempty (support f)
case mp.intro.mk.inr f : Equiv.Perm β„• a b : β„• hab : (a, b) ∈ Inversions f hb : b ∈ support f ⊒ Set.Nonempty (support f)
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 β„• a b : β„• hab : (a, b) ∈ Inversions f hb : b ∈ support f ⊒ Set.Nonempty (support f)
no goals
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 β„• a b : β„• hab : (a, b) ∈ Inversions f ha : a ∈ support f ⊒ Set.Nonempty (support f)
no goals
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 β„• a b : β„• hab : (a, b) ∈ Inversions f hb : b ∈ support f ⊒ Set.Nonempty (support f)
no goals
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 β„• ⊒ Set.Nonempty (support f) β†’ Set.Nonempty (Inversions f)
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
inversions_comp
[230, 1]
[241, 67]
ext ⟨a, b⟩
f g : Equiv.Perm β„• ⊒ Inversions (f * g) = (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) βˆ† Inversions g
case h.mk f g : Equiv.Perm β„• a b : β„• ⊒ (a, b) ∈ Inversions (f * g) ↔ (a, b) ∈ (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) βˆ† Inversions g
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 β„• a b : β„• ⊒ (a, b) ∈ Inversions (f * g) ↔ (a, b) ∈ (⇑(Equiv.prodCongr g g) ⁻¹' Inversions f) βˆ† Inversions g
case h.mk f g : Equiv.Perm β„• a b : β„• ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case h.mk.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f f g : Equiv.Perm β„• a b : β„• h : a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f f g : Equiv.Perm β„• a b : β„• h : a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
f g : Equiv.Perm β„• a b : β„• h : a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• h : a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case inl f g : Equiv.Perm β„• a b : β„• h : a < b hgab : g a < g b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f case inr f g : Equiv.Perm β„• a b : β„• h : a < b hgba : g b ≀ g a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• h : a < b hgab : g a < g b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f case inr f g : Equiv.Perm β„• a b : β„• h : a < b hgba : g b ≀ g a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case inr f g : Equiv.Perm β„• a b : β„• h : a < b hgba : g b ≀ g a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• h : a < b hgba : g b ≀ g a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
no goals
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case h.mk.inr.inl f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b = a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f case h.mk.inr.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b = a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f case h.mk.inr.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case h.mk.inr.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
no goals
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b = a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
no goals
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case h.mk.inr.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (b, a) ∈ Inversions (f * g) ↔ (g b, g a) ∈ Inversions f ∧ (b, a) βˆ‰ Inversions g ∨ (b, a) ∈ Inversions g ∧ (g b, g a) βˆ‰ Inversions f
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (b, a) ∈ Inversions (f * g) ↔ (g b, g a) ∈ Inversions f ∧ (b, a) βˆ‰ Inversions g ∨ (b, a) ∈ Inversions g ∧ (g b, g a) βˆ‰ Inversions f
no goals
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 β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
case h.mk.inr.inr f g : Equiv.Perm β„• a b : β„• this : βˆ€ (f g : Equiv.Perm β„•) (a b : β„•), a < b β†’ ((a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f) h : Β¬a < b h' : b < a ⊒ (b, a) ∈ Inversions (f * g) ↔ (g b, g a) ∈ Inversions f ∧ (b, a) βˆ‰ Inversions g ∨ (b, a) ∈ Inversions g ∧ (g b, g a) βˆ‰ Inversions f
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 β„• a b : β„• h : a < b hgab : g a < g b ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
no goals
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 β„• a b : β„• h : a < b hgba : g b ≀ g a ⊒ (a, b) ∈ Inversions (f * g) ↔ (g a, g b) ∈ Inversions f ∧ (a, b) βˆ‰ Inversions g ∨ (a, b) ∈ Inversions g ∧ (g a, g b) βˆ‰ Inversions f
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions
[249, 1]
[250, 70]
rfl
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± p : Ξ± Γ— Ξ± ⊒ p ∈ StdInversions g ↔ p.1 < p.2 ∧ g p.2 ≀ g p.1
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions'
[252, 1]
[253, 59]
rfl
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a, b) ∈ StdInversions g ↔ a < b ∧ g b ≀ g a
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
rw [mem_stdinversions']
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a, b) ∈ StdInversions g ↔ a < b ∧ g b < g a
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
by_cases h : a = b
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
case pos Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
. simp [h]
case pos Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
. have h' : g a β‰  g b := fun hg => h (g.injective hg) simp [h, h'.symm.le_iff_lt]
case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
simp [h]
case pos Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
mem_stdinversions''
[255, 1]
[261, 32]
have h' : g a β‰  g b := fun hg => h (g.injective hg)
case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
case neg Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b h' : g a β‰  g b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
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 Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± h : Β¬a = b h' : g a β‰  g b ⊒ a < b ∧ g b ≀ g a ↔ a < b ∧ g b < g a
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
not_mem_stdinversions_diag
[263, 1]
[264, 59]
simp [mem_stdinversions']
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a : Ξ± ⊒ (a, a) βˆ‰ StdInversions g
no goals
https://github.com/mguaypaq/lean-bruhat.git
1666a1bee2b520d5ba8a662310b4bd257fcf7ac2
Bruhat/Word.lean
stdinversions_inversions
[267, 1]
[278, 65]
ext ⟨a, b⟩
Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± ⊒ Inversions g = StdInversions g βˆͺ Prod.swap ⁻¹' StdInversions g
case h.mk Ξ± : Type u_1 inst✝ : LinearOrder Ξ± g : Equiv.Perm Ξ± a b : Ξ± ⊒ (a, b) ∈ Inversions g ↔ (a, b) ∈ StdInversions g βˆͺ Prod.swap ⁻¹' StdInversions g