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stringclasses 147
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stringlengths 7
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stringlengths 1
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stringlengths 6
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https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | all_goals clear s hs | case H0
s : Set β
hs : Set.Finite s
β’ RepbleByWord_aux β
case H1
s : Set β
hs : Set.Finite s
β’ β {a : β} {s : Set β}, a β s β Set.Finite s β RepbleByWord_aux s β RepbleByWord_aux (insert a s) | case H0
β’ RepbleByWord_aux β
case H1
β’ β {a : β} {s : Set β}, a β 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 β¨_, idHasWordβ© | case H0
β’ RepbleByWord_aux β
case H1
β’ β {a : β} {s : Set β}, a β s β Set.Finite s β RepbleByWord_aux s β RepbleByWord_aux (insert a s) | case H1
β’ β {a : β} {s : Set β}, a β 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 β s
obtain β¨w, hwβ© := hs_repble g hgs
obtain β¨w_swap, hw_swapβ© := 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
β’ β {a : β} {s : Set β}, a β 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
β’ β {a : β} {s : Set β}, a β s β Set.Finite s β RepbleByWord_aux s β RepbleByWord_aux (insert a s) | case H1
β’ β {a : β} {s : Set β}, a β 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
β’ RepbleByWord_aux β
| case H0
f : Equiv.Perm β
hf : support f β β
β’ 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 β β
β’ RepbleByWord f | case H0
f : Equiv.Perm β
hf : f = 1
β’ 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
β’ RepbleByWord f | case H0
β’ RepbleByWord 1 |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | exact β¨_, idHasWordβ© | case H0
β’ 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
β’ β {a : β} {s : Set β}, a β s β Set.Finite s β RepbleByWord_aux s β RepbleByWord_aux (insert a s) | case H1
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
β’ 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
β’ RepbleByWord f | case H1
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ RepbleByWord f |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | suffices hgs : support g β s | case H1
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ RepbleByWord f | case H1
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | obtain β¨w, hwβ© := hs_repble g hgs | case H1
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s | case H1.intro
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | repbleByWord_aux | [130, 1] | [150, 7] | obtain β¨w_swap, hw_swapβ© := swapHasWord a (f a) | case H1.intro
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s | case H1.intro.intro
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
β’ RepbleByWord f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s | case h
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
β’ Word.toPerm (w_swap ++ w) = f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hgs : support g β s
w : Word
hw : Word.toPerm w = g
w_swap : Word
hw_swap : Word.toPerm w_swap = Equiv.swap a (f a)
β’ Word.toPerm (w_swap ++ w) = f
case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s | case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
β’ support g β s | case hgs
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
β’ b β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
β’ b β s | case pos
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : b = a
β’ b β s
case neg
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : Β¬b = a
β’ b β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : b = a
β’ b β s
case neg
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : Β¬b = a
β’ b β s | case neg
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : Β¬b = a
β’ b β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : Β¬b = a
β’ b β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β 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 β support g
h : b = a
β’ b β s | case pos
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a β support g
β’ a β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a β support g
β’ a β s | case pos
a : β
s : Set β
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a β support g
β’ a β 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
aβΒΉ : a β s
aβ : Set.Finite s
hs_repble : RepbleByWord_aux s
f : Equiv.Perm β
hf : support f β insert a s
g : Equiv.Perm β := Equiv.swap a (f a) * f
hg : g = Equiv.swap a (f a) * f
hb : a β support g
β’ a β 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)
β’ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
p : Ξ± Γ Ξ±
β’ p β Inversions g β (p.1 < p.2 β g p.2 β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (a < b β g b β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a : Ξ±
β’ (a, a) β 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (b, a) β Inversions g | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b) | case inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b)
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b)
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b) | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b) | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
h' : g a β g b
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a < b
h' : g a β g b
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b) | case inr.inl
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b = a
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b)
case inr.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b < a
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b = a
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b)
case inr.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b < a
β’ (a < b β g b β€ g a) β (b < a β g a β€ g b) | case inr.inr
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b < a
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b < a
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b = a
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
this :
β {Ξ± : Type u_1} [inst : LinearOrder Ξ±] (g : Equiv.Perm Ξ±) (a b : Ξ±),
a < b β ((a < b β g b β€ g a) β (b < a β g a β€ g b))
h : Β¬a < b
h' : b < a
β’ (a < b β g b β€ g a) β (b < a β g a β€ 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 β’ | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : (a, b) β Inversions g
β’ a β support g β¨ b β support g | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a < b β g b β€ g a
β’ Β¬g a = a β¨ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a < b β g b β€ g a
β’ Β¬g a = a β¨ Β¬g b = b | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a < b β g b β€ g a
hab' : g a = a β§ g b = b
β’ 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, β not_le] at hab | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : a < b β g b β€ g a
hab' : g a = a β§ g b = b
β’ False | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬b β€ a β b β€ a
hab' : g a = a β§ g b = b
β’ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
hab : Β¬b β€ a β b β€ a
hab' : g a = a β§ g b = b
β’ 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 : β n, f n β n
β’ (Nat.find hf, fβ»ΒΉ (Nat.find hf)) β Inversions f | f : Equiv.Perm β
hf : β n, f n β n
β’ Nat.find hf < fβ»ΒΉ (Nat.find hf) β f (fβ»ΒΉ (Nat.find hf)) β€ 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 : β n, f n β n
β’ Nat.find hf < fβ»ΒΉ (Nat.find hf) β f (fβ»ΒΉ (Nat.find hf)) β€ 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 : β m < n, n β€ f m | f : Equiv.Perm β
n : β
hn : f n < n
β’ β m < n, (m, n) β Inversions f | f : Equiv.Perm β
n : β
hn : f n < n
this : β m < n, n β€ f m
β’ β m < n, (m, n) β Inversions f
case this
f : Equiv.Perm β
n : β
hn : f n < n
β’ β m < n, n β€ f m |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | . obtain β¨m, β¨hmn, hm'β©β© := this
refine' β¨m, hmn, _β©
simp [mem_inversions', hmn, (hn.trans_le hm').le] | f : Equiv.Perm β
n : β
hn : f n < n
this : β m < n, n β€ f m
β’ β m < n, (m, n) β Inversions f
case this
f : Equiv.Perm β
n : β
hn : f n < n
β’ β m < n, n β€ f m | case this
f : Equiv.Perm β
n : β
hn : f n < n
β’ β m < n, n β€ 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
β’ β m < n, n β€ f m | case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
β’ 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)) β Finset.range n | case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
β’ False | case key1
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
β’ Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
key1 : Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
β’ 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 β’
obtain β¨m', β¨hm', rflβ©β© := 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 : β m < n, f m < n
β’ Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
key1 : Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
β’ False | case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
key1 : Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
β’ 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 : β m < n, f m < n
key1 : Finset.image (βf) (Finset.range (n + 1)) β Finset.range n
β’ False | case this
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
key1 : Finset.card (Finset.image (βf) (Finset.range (n + 1))) β€ Finset.card (Finset.range n)
β’ 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 : β m < n, f m < n
key1 : Finset.card (Finset.image (βf) (Finset.range (n + 1))) β€ Finset.card (Finset.range n)
β’ False | no goals |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | obtain β¨m, β¨hmn, hm'β©β© := this | f : Equiv.Perm β
n : β
hn : f n < n
this : β m < n, n β€ f m
β’ β m < n, (m, n) β Inversions f | case intro.intro
f : Equiv.Perm β
n : β
hn : f n < n
m : β
hmn : m < n
hm' : n β€ f m
β’ β m < n, (m, n) β Inversions f |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | refine' β¨m, hmn, _β© | case intro.intro
f : Equiv.Perm β
n : β
hn : f n < n
m : β
hmn : m < n
hm' : n β€ f m
β’ β m < n, (m, n) β Inversions f | case intro.intro
f : Equiv.Perm β
n : β
hn : f n < n
m : β
hmn : m < n
hm' : n β€ f m
β’ (m, n) β 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 β€ f m
β’ (m, n) β 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 : β m < n, f m < n
β’ Finset.image (βf) (Finset.range (n + 1)) β Finset.range n | case key1
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
aβ : β
hm : aβ β Finset.image (βf) (Finset.range (n + 1))
β’ aβ β 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 β’ | case key1
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
aβ : β
hm : aβ β Finset.image (βf) (Finset.range (n + 1))
β’ aβ β Finset.range n | case key1
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
aβ : β
hm : β a < n + 1, f a = aβ
β’ aβ < n |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | exists_inversion_of_apply_lt | [197, 1] | [212, 81] | obtain β¨m', β¨hm', rflβ©β© := hm | case key1
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
aβ : β
hm : β a < n + 1, f a = aβ
β’ aβ < n | case key1.intro.intro
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
m' : β
hm' : m' < n + 1
β’ 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 : β m < n, f m < n
m' : β
hm' : m' < n + 1
β’ f m' < n | case key1.intro.intro.inl
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' < n
β’ f m' < n
case key1.intro.intro.inr
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' = n
β’ 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 : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' < n
β’ f m' < n
case key1.intro.intro.inr
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' = n
β’ f m' < n | case key1.intro.intro.inr
f : Equiv.Perm β
n : β
hn : f n < n
h_lt : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' = n
β’ 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 : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' = n
β’ 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 : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' < n
β’ 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 : β m < n, f m < n
m' : β
hm'β : m' < n + 1
hm' : m' = n
β’ 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 [β not_iff_not] | f : Equiv.Perm β
β’ Inversions f = β
β support f = β
| f : Equiv.Perm β
β’ Β¬Inversions f = β
β Β¬support f = β
|
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’ Β¬Inversions f = β
β Β¬support f = β
| f : Equiv.Perm β
β’ Set.Nonempty (Inversions f) β 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
β’ Set.Nonempty (Inversions f) β Set.Nonempty (support f) | case mp
f : Equiv.Perm β
β’ Set.Nonempty (Inversions f) β Set.Nonempty (support f)
case mpr
f : Equiv.Perm β
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 β¨β¨a, bβ©, habβ©
cases' mem_support_of_inversion _ _ _ hab with ha hb
. exact β¨_, haβ©
. exact β¨_, hbβ© | case mp
f : Equiv.Perm β
β’ Set.Nonempty (Inversions f) β Set.Nonempty (support f)
case mpr
f : Equiv.Perm β
β’ Set.Nonempty (support f) β Set.Nonempty (Inversions f) | case mpr
f : Equiv.Perm β
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 => β¨_, min_support_mem_inversions f hfβ© | case mpr
f : Equiv.Perm β
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 β¨β¨a, bβ©, habβ© | case mp
f : Equiv.Perm β
β’ Set.Nonempty (Inversions f) β Set.Nonempty (support f) | case mp.intro.mk
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
β’ 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) β Inversions f
β’ Set.Nonempty (support f) | case mp.intro.mk.inl
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
ha : a β support f
β’ Set.Nonempty (support f)
case mp.intro.mk.inr
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
hb : b β support f
β’ 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 β¨_, haβ© | case mp.intro.mk.inl
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
ha : a β support f
β’ Set.Nonempty (support f)
case mp.intro.mk.inr
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
hb : b β support f
β’ Set.Nonempty (support f) | case mp.intro.mk.inr
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
hb : b β support f
β’ 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 β¨_, hbβ© | case mp.intro.mk.inr
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
hb : b β support f
β’ 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 β¨_, haβ© | case mp.intro.mk.inl
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
ha : a β support f
β’ 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 β¨_, hbβ© | case mp.intro.mk.inr
f : Equiv.Perm β
a b : β
hab : (a, b) β Inversions f
hb : b β support f
β’ 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 => β¨_, min_support_mem_inversions f hfβ© | case mpr
f : Equiv.Perm β
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 β¨a, bβ© | f g : Equiv.Perm β
β’ Inversions (f * g) = (β(Equiv.prodCongr g g) β»ΒΉ' Inversions f) β Inversions g | case h.mk
f g : Equiv.Perm β
a b : β
β’ (a, b) β Inversions (f * g) β (a, b) β (β(Equiv.prodCongr g g) β»ΒΉ' Inversions f) β 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
β’ (a, b) β Inversions (f * g) β (a, b) β (β(Equiv.prodCongr g g) β»ΒΉ' Inversions f) β Inversions g | case h.mk
f g : Equiv.Perm β
a b : β
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case h.mk.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
f g : Equiv.Perm β
a b : β
h : a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
f g : Equiv.Perm β
a b : β
h : a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | f g : Equiv.Perm β
a b : β
h : a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case inl
f g : Equiv.Perm β
a b : β
h : a < b
hgab : g a < g b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
case inr
f g : Equiv.Perm β
a b : β
h : a < b
hgba : g b β€ g a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
case inr
f g : Equiv.Perm β
a b : β
h : a < b
hgba : g b β€ g a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case inr
f g : Equiv.Perm β
a b : β
h : a < b
hgba : g b β€ g a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 β€ g a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case h.mk.inr.inl
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b = a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
case h.mk.inr.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b = a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f
case h.mk.inr.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case h.mk.inr.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b = a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case h.mk.inr.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (b, a) β Inversions (f * g) β
(g b, g a) β Inversions f β§ (b, a) β Inversions g β¨ (b, a) β Inversions g β§ (g b, g a) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (b, a) β Inversions (f * g) β
(g b, g a) β Inversions f β§ (b, a) β Inversions g β¨ (b, a) β Inversions g β§ (g b, g a) β 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 :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f | case h.mk.inr.inr
f g : Equiv.Perm β
a b : β
this :
β (f g : Equiv.Perm β) (a b : β),
a < b β
((a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β Inversions f)
h : Β¬a < b
h' : b < a
β’ (b, a) β Inversions (f * g) β
(g b, g a) β Inversions f β§ (b, a) β Inversions g β¨ (b, a) β Inversions g β§ (g b, g a) β 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
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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 β€ g a
β’ (a, b) β Inversions (f * g) β
(g a, g b) β Inversions f β§ (a, b) β Inversions g β¨ (a, b) β Inversions g β§ (g a, g b) β 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
p : Ξ± Γ Ξ±
β’ p β StdInversions g β p.1 < p.2 β§ g p.2 β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β StdInversions g β a < b β§ g b β€ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β StdInversions g β a < b β§ g b < g a | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ a < b β§ g b β€ g a β a < b β§ g b < g a | case pos
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a = b
β’ a < b β§ g b β€ g a β a < b β§ g b < g a
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a = b
β’ a < b β§ g b β€ g a β a < b β§ g b < g a
case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
β’ a < b β§ g b β€ g a β a < b β§ g b < g a | case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : a = b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
β’ a < b β§ g b β€ g a β a < b β§ g b < g a | case neg
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
h' : g a β g b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
h : Β¬a = b
h' : g a β g b
β’ a < b β§ g b β€ g a β a < b β§ 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
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a : Ξ±
β’ (a, a) β StdInversions g | no goals |
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Word.lean | stdinversions_inversions | [267, 1] | [278, 65] | ext β¨a, bβ© | Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
β’ Inversions g = StdInversions g βͺ Prod.swap β»ΒΉ' StdInversions g | case h.mk
Ξ± : Type u_1
instβ : LinearOrder Ξ±
g : Equiv.Perm Ξ±
a b : Ξ±
β’ (a, b) β Inversions g β (a, b) β StdInversions g βͺ Prod.swap β»ΒΉ' StdInversions g |
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