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stringclasses 147
values | file_path
stringlengths 7
101
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stringlengths 1
94
| start
stringlengths 6
10
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stringlengths 6
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stringlengths 1
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stringlengths 73
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | rw [mul_support_eq, mul_support_eq] | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y * swap y z * swap x y).support ⊆ {x, y, z} | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => (swap x y * swap y z) ((swap x y) x_1) ≠ x_1)
(filter (fun x_1 => (swap x y) ((swap y z) x_1) ≠ x_1) ((swap x y).support ∪ (swap y z).support) ∪
(swap x y).support) ⊆
{x, y, z} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x y * swap y z * swap x y).support ⊆ {x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | simp only [mul_apply, ne_eq, swap_support, filter_congr_decidable, if_neg hxy, if_neg hyz] | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => (swap x y * swap y z) ((swap x y) x_1) ≠ x_1)
(filter (fun x_1 => (swap x y) ((swap y z) x_1) ≠ x_1) ((swap x y).support ∪ (swap y z).support) ∪
(swap x y).support) ⊆
{x, y, z} | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => ¬(swap x y) ((swap y z) ((swap x y) x_1)) = x_1)
(filter (fun x_1 => ¬(swap x y) ((swap y z) x_1) = x_1) ({x, y} ∪ {y, z}) ∪ {x, y}) ⊆
{x, y, z} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => (swap x y * swap y z) ((swap x y) x_1) ≠ x_1)
(filter (fun x_1 => (swap x y) ((swap y z) x_1) ≠ x_1) ((swap x y).support ∪ (swap y z).support) ∪
(swap x y).support) ⊆
{x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | refine (filter_subset _ _).trans (union_subset ((filter_subset _ _).trans ?_) (by aesop)) | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => ¬(swap x y) ((swap y z) ((swap x y) x_1)) = x_1)
(filter (fun x_1 => ¬(swap x y) ((swap y z) x_1) = x_1) ({x, y} ∪ {y, z}) ∪ {x, y}) ⊆
{x, y, z} | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ {x, y} ∪ {y, z} ⊆ {x, y, z} | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ filter (fun x_1 => ¬(swap x y) ((swap y z) ((swap x y) x_1)) = x_1)
(filter (fun x_1 => ¬(swap x y) ((swap y z) x_1) = x_1) ({x, y} ∪ {y, z}) ∪ {x, y}) ⊆
{x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | apply union_subset <;> aesop | case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ {x, y} ∪ {y, z} ⊆ {x, y, z} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hf
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ {x, y} ∪ {y, z} ⊆ {x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | aesop | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ {x, y} ⊆ {x, y, z} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ {x, y} ⊆ {x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swap_conj_eq | [272, 1] | [286, 7] | aesop | case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x z).support ⊆ {x, y, z} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.hg
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
z : α
hxz : x ≠ z
hyz : y ≠ z
hxy : x ≠ y
⊢ (swap x z).support ⊆ {x, y, z}
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | intro y hy | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
⊢ (f * swap x (f⁻¹ x)).support ⊆ erase f.support x | α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ y ∈ erase f.support x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
⊢ (f * swap x (f⁻¹ x)).support ⊆ erase f.support x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | simp only [mem_erase, ne_eq, mem_support_iff] | α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ y ∈ erase f.support x | α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ ¬y = x ∧ ¬f y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ y ∈ erase f.support x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | obtain (rfl | hne) := eq_or_ne y x | α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ ¬y = x ∧ ¬f y = y | case inl
α : Type u_1
y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
y : α
hx : y ∈ f.support
hy : y ∈ (f * swap y (f⁻¹ y)).support
⊢ ¬y = y ∧ ¬f y = y
case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
⊢ ¬y = x ∧ ¬f y = y | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
⊢ ¬y = x ∧ ¬f y = y
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | refine ⟨hne, fun hy' ↦ ?_⟩ | case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
⊢ ¬y = x ∧ ¬f y = y | case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
hy' : f y = y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
⊢ ¬y = x ∧ ¬f y = y
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | simp only [mem_support_iff, mul_apply, ne_eq] at hy | case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
hy' : f y = y
⊢ False | case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hy : y ∈ (f * swap x (f⁻¹ x)).support
hne : y ≠ x
hy' : f y = y
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | obtain (rfl | hne') := eq_or_ne y (f⁻¹ x) | case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
⊢ False | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy' : f (f⁻¹ x) = f⁻¹ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
⊢ False
case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
hne' : y ≠ f⁻¹ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | rw [swap_apply_of_ne_of_ne hne hne'] at hy | case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
hne' : y ≠ f⁻¹ x
⊢ False | case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f y = y
hne' : y ≠ f⁻¹ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f ((swap x (f⁻¹ x)) y) = y
hne' : y ≠ f⁻¹ x
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | exact hy hy' | case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f y = y
hne' : y ≠ f⁻¹ x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
y : α
hne : y ≠ x
hy' : f y = y
hy : ¬f y = y
hne' : y ≠ f⁻¹ x
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | simp at hy | case inl
α : Type u_1
y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
y : α
hx : y ∈ f.support
hy : y ∈ (f * swap y (f⁻¹ y)).support
⊢ ¬y = y ∧ ¬f y = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
y : α
hx : y ∈ f.support
hy : y ∈ (f * swap y (f⁻¹ y)).support
⊢ ¬y = y ∧ ¬f y = y
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | simp only [swap_apply_right, apply_inv_apply] at hy' | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy' : f (f⁻¹ x) = f⁻¹ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
⊢ False | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy' : f (f⁻¹ x) = f⁻¹ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | rw [← hy'] at hne | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : f⁻¹ x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_mul_pair_subset | [288, 1] | [301, 15] | exact hne rfl | case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
hx : x ∈ f.support
hne : x ≠ x
hy : ¬f ((swap x (f⁻¹ x)) (f⁻¹ x)) = f⁻¹ x
hy' : x = f⁻¹ x
⊢ False
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swapsOf_univ_eq | [310, 9] | [312, 24] | simp [swaps, swapsOf] | α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ swapsOf Set.univ = swaps α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ swapsOf Set.univ = swaps α
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swapsOf_support_subset | [314, 1] | [317, 48] | obtain ⟨i,j, hi, hj, hne, rfl⟩ := hf | α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
hf : f ∈ swapsOf t
⊢ ↑f.support ⊆ t | case intro.intro.intro.intro.intro
α : Type u_1
x y : α
g : Finperm α
inst✝ : DecidableEq α
t : Set α
i j : α
hi : i ∈ t
hj : j ∈ t
hne : i ≠ j
⊢ ↑(swap i j).support ⊆ t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
hf : f ∈ swapsOf t
⊢ ↑f.support ⊆ t
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swapsOf_support_subset | [314, 1] | [317, 48] | rwa [swap_support_of_ne hne, coe_insert, Set.insert_subset_iff, coe_singleton,
Set.singleton_subset_iff, and_iff_right hi] | case intro.intro.intro.intro.intro
α : Type u_1
x y : α
g : Finperm α
inst✝ : DecidableEq α
t : Set α
i j : α
hi : i ∈ t
hj : j ∈ t
hne : i ≠ j
⊢ ↑(swap i j).support ⊆ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type u_1
x y : α
g : Finperm α
inst✝ : DecidableEq α
t : Set α
i j : α
hi : i ∈ t
hj : j ∈ t
hne : i ≠ j
⊢ ↑(swap i j).support ⊆ t
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | obtain (h | h) := eq_or_ne f.support ∅ | α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
⊢ f ∈ Subgroup.closure (swapsOf t) | case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ f ∈ Subgroup.closure (swapsOf t)
case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support ≠ ∅
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | simp only [ne_eq, eq_empty_iff_forall_not_mem, mem_support_iff, not_not, not_forall] at h | case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support ≠ ∅
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : ∃ x, ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support ≠ ∅
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | obtain ⟨x, hx⟩ := h | case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : ∃ x, ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : ∃ x, ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hx' : x ∈ f.support := by simpa | case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | set g := f * swap x (f⁻¹ x) with hg_def | case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hsupp : g.support ⊆ _ := support_mul_pair_subset hx' | case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have _ : g.support.card < f.support.card := by
exact card_lt_card <| (hsupp.trans_ssubset (erase_ssubset hx')) | case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hg_supp : (g.support : Set α) ⊆ t | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ f ∈ Subgroup.closure (swapsOf t) | case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ ↑g.support ⊆ t
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hg := support_closure_aux hg_supp | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t) | case hs
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | have hf' : f = g * (swap x (f⁻¹ x)) | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t) | case hf'
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f = g * swap x (f⁻¹ x)
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | rw [hf'] | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t) | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ g * swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | exact Subgroup.mul_mem _ hg hs | case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ g * swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
hf' : f = g * swap x (f⁻¹ x)
⊢ g * swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | rw [show f = 1 from (support_eq_empty_iff _).1 h] | case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ f ∈ Subgroup.closure (swapsOf t) | case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ 1 ∈ Subgroup.closure (swapsOf t) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ f ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | exact Subgroup.one_mem _ | case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ 1 ∈ Subgroup.closure (swapsOf t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
x y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
h : f.support = ∅
⊢ 1 ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | simpa | α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
⊢ x ∈ f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
⊢ x ∈ f.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | exact card_lt_card <| (hsupp.trans_ssubset (erase_ssubset hx')) | α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
⊢ card g.support < card f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
⊢ card g.support < card f.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | refine (Finset.coe_subset.2 hsupp).trans (subset_trans (Finset.coe_subset.2 ?_) hf) | case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ ↑g.support ⊆ t | case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ erase f.support x ⊆ f.support | Please generate a tactic in lean4 to solve the state.
STATE:
case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ ↑g.support ⊆ t
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | exact erase_subset x f.support | case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ erase f.support x ⊆ f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hg_supp
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
⊢ erase f.support x ⊆ f.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | refine Subgroup.subset_closure ⟨_, _, hf hx', hf ?_, ?_, rfl⟩ | case hs
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t) | case hs.refine_1
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f⁻¹ x ∈ ↑f.support
case hs.refine_2
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ x ≠ f⁻¹ x | Please generate a tactic in lean4 to solve the state.
STATE:
case hs
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | rwa [ne_eq, eq_inv_iff_eq] | case hs.refine_2
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ x ≠ f⁻¹ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.refine_2
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ x ≠ f⁻¹ x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | rwa [mem_coe, mem_support_iff, apply_inv_apply, ne_eq, eq_inv_iff_eq] | case hs.refine_1
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f⁻¹ x ∈ ↑f.support | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hs.refine_1
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
⊢ f⁻¹ x ∈ ↑f.support
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.support_closure_aux | [322, 1] | [346, 35] | rw [hg_def, mul_assoc, swap_mul_swap, mul_one] | case hf'
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f = g * swap x (f⁻¹ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hf'
α : Type u_1
x✝¹ y : α
f✝ g✝ : Finperm α
inst✝ : DecidableEq α
t : Set α
f : Finperm α
hf : ↑f.support ⊆ t
x : α
hx : ¬f x = x
hx' : x ∈ f.support
g : Finperm α := f * swap x (f⁻¹ x)
hg_def : g = f * swap x (f⁻¹ x)
hsupp : g.support ⊆ erase f.support x
x✝ : card g.support < card f.support
hg_supp : ↑g.support ⊆ t
hg : g ∈ Subgroup.closure (swapsOf t)
hs : swap x (f⁻¹ x) ∈ Subgroup.closure (swapsOf t)
⊢ f = g * swap x (f⁻¹ x)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swaps_eq_top | [351, 1] | [353, 24] | rw [← swapsOf_univ_eq, ← restrict_univ] | α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ Subgroup.closure (swaps α) = ⊤ | α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ Subgroup.closure (swapsOf Set.univ) = restrict Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ Subgroup.closure (swaps α) = ⊤
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swaps_eq_top | [351, 1] | [353, 24] | exact cl_swapsOf_eq _ | α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ Subgroup.closure (swapsOf Set.univ) = restrict Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x y : α✝
f g : Finperm α✝
inst✝¹ : DecidableEq α✝
α : Type u_2
inst✝ : DecidableEq α
⊢ Subgroup.closure (swapsOf Set.univ) = restrict Set.univ
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swapsAt_subset_swapsOf | [358, 1] | [360, 59] | rintro s ⟨i, hi, rfl⟩ | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ swapsAt x t ⊆ swapsOf (insert x t) | case intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
i : α
hi : i ∈ t \ {x}
⊢ (fun x_1 => swap x x_1) i ∈ swapsOf (insert x t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ swapsAt x t ⊆ swapsOf (insert x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.swapsAt_subset_swapsOf | [358, 1] | [360, 59] | exact ⟨x, _, Or.inl rfl, Or.inr hi.1, Ne.symm hi.2, rfl⟩ | case intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
i : α
hi : i ∈ t \ {x}
⊢ (fun x_1 => swap x x_1) i ∈ swapsOf (insert x t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
i : α
hi : i ∈ t \ {x}
⊢ (fun x_1 => swap x x_1) i ∈ swapsOf (insert x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | have aux : ∀ {y}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t) | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ Subgroup.closure (swapsAt x t) = restrict (insert x t) | case aux
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) = restrict (insert x t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ Subgroup.closure (swapsAt x t) = restrict (insert x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rw [← cl_swapsOf_eq, le_antisymm_iff] | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) = restrict (insert x t) | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) ≤ Subgroup.closure (swapsOf (insert x t)) ∧
Subgroup.closure (swapsOf (insert x t)) ≤ Subgroup.closure (swapsAt x t) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) = restrict (insert x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | refine ⟨Subgroup.closure_mono (swapsAt_subset_swapsOf _ _), (Subgroup.closure_le _).2 ?_⟩ | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) ≤ Subgroup.closure (swapsOf (insert x t)) ∧
Subgroup.closure (swapsOf (insert x t)) ≤ Subgroup.closure (swapsAt x t) | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ swapsOf (insert x t) ⊆ ↑(Subgroup.closure (swapsAt x t)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ Subgroup.closure (swapsAt x t) ≤ Subgroup.closure (swapsOf (insert x t)) ∧
Subgroup.closure (swapsOf (insert x t)) ≤ Subgroup.closure (swapsAt x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rintro _ ⟨i, j, hi, hj, hne, rfl⟩ | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ swapsOf (insert x t) ⊆ ↑(Subgroup.closure (swapsAt x t)) | case intro.intro.intro.intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
⊢ swapsOf (insert x t) ⊆ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | obtain (rfl | hjne) := eq_or_ne x j | case intro.intro.intro.intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t)) | case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap i x ∈ ↑(Subgroup.closure (swapsAt x t))
case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rw [← swap_conj_eq hne hjne, swap_comm i] | case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t)) | case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap x i * swap x j * swap x i ∈ ↑(Subgroup.closure (swapsAt x t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap i j ∈ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | exact mul_mem (mul_mem (aux hi) (aux hj)) (aux hi) | case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap x i * swap x j * swap x i ∈ ↑(Subgroup.closure (swapsAt x t)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i j : α
hi : i ∈ insert x t
hj : j ∈ insert x t
hne : i ≠ j
hjne : x ≠ j
⊢ swap x i * swap x j * swap x i ∈ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | intro y hy | case aux
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t) | case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x t
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | Please generate a tactic in lean4 to solve the state.
STATE:
case aux
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
⊢ ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rw [← Set.insert_diff_singleton] at hy | case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x t
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x (t \ {x})
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | Please generate a tactic in lean4 to solve the state.
STATE:
case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x t
⊢ swap x y ∈ Subgroup.closure (swapsAt x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | obtain (rfl | hy) := hy | case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x (t \ {x})
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ swap y y ∈ Subgroup.closure (swapsAt y t)
case aux.inr
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ t \ {x}
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | Please generate a tactic in lean4 to solve the state.
STATE:
case aux
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ insert x (t \ {x})
⊢ swap x y ∈ Subgroup.closure (swapsAt x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | exact Subgroup.subset_closure ⟨y, hy, rfl⟩ | case aux.inr
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ t \ {x}
⊢ swap x y ∈ Subgroup.closure (swapsAt x t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case aux.inr
α : Type u_1
x✝ y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
y : α
hy : y ∈ t \ {x}
⊢ swap x y ∈ Subgroup.closure (swapsAt x t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rw [swap_self, ← one_def] | case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ swap y y ∈ Subgroup.closure (swapsAt y t) | case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ 1 ∈ Subgroup.closure (swapsAt y t) | Please generate a tactic in lean4 to solve the state.
STATE:
case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ swap y y ∈ Subgroup.closure (swapsAt y t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | exact Subgroup.one_mem _ | case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ 1 ∈ Subgroup.closure (swapsAt y t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case aux.inl
α : Type u_1
x y✝ : α
f g : Finperm α
inst✝ : DecidableEq α
t : Set α
y : α
⊢ 1 ∈ Subgroup.closure (swapsAt y t)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | rw [swap_comm] | case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap i x ∈ ↑(Subgroup.closure (swapsAt x t)) | case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap x i ∈ ↑(Subgroup.closure (swapsAt x t)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap i x ∈ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq | [362, 1] | [377, 53] | exact aux hi | case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap x i ∈ ↑(Subgroup.closure (swapsAt x t)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
aux : ∀ {y : α}, y ∈ insert x t → swap x y ∈ Subgroup.closure (swapsAt x t)
i : α
hi : i ∈ insert x t
hj : x ∈ insert x t
hne : i ≠ x
⊢ swap x i ∈ ↑(Subgroup.closure (swapsAt x t))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.cl_swapsAt_eq' | [379, 1] | [381, 47] | rw [cl_swapsAt_eq, Set.insert_eq_of_mem hxt] | α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
hxt : x ∈ t
⊢ Subgroup.closure (swapsAt x t) = restrict t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x✝ y : α
f g : Finperm α
inst✝ : DecidableEq α
x : α
t : Set α
hxt : x ∈ t
⊢ Subgroup.closure (swapsAt x t) = restrict t
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | induction' n with n IH | α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow n) | case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ swap 0 Nat.zero ∈ Subgroup.closure (adjSwapsBelow Nat.zero)
case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (Nat.succ n) ∈ Subgroup.closure (adjSwapsBelow (Nat.succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | simp [Nat.succ_eq_add_one] | case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (Nat.succ n) ∈ Subgroup.closure (adjSwapsBelow (Nat.succ n)) | case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (Nat.succ n) ∈ Subgroup.closure (adjSwapsBelow (Nat.succ n))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | rw [← swap_conj_eq (x := 0) (y := n) (z := n+1)] | case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n * swap n (n + 1) * swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1))
case succ.hxz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ 0 ≠ n + 1
case succ.hyz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ n ≠ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Ne.symm (Nat.succ_ne_self n) | case succ.hyz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ n ≠ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hyz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ n ≠ n + 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | simp only [Nat.zero_eq, swap_self, ← one_def] | case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ swap 0 Nat.zero ∈ Subgroup.closure (adjSwapsBelow Nat.zero) | case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ 1 ∈ Subgroup.closure (adjSwapsBelow 0) | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ swap 0 Nat.zero ∈ Subgroup.closure (adjSwapsBelow Nat.zero)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Subgroup.one_mem (Subgroup.closure (adjSwapsBelow 0)) | case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ 1 ∈ Subgroup.closure (adjSwapsBelow 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ 1 ∈ Subgroup.closure (adjSwapsBelow 0)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | refine Subgroup.mul_mem _ (Subgroup.mul_mem _ ?_ ?_) ?_ | case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n * swap n (n + 1) * swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | case succ.refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1))
case succ.refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap n (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1))
case succ.refine_3
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n * swap n (n + 1) * swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Subgroup.closure_mono (adjSwapsBelow_mono (le_self_add : n ≤ n +1)) IH | case succ.refine_3
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.refine_3
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Subgroup.closure_mono (adjSwapsBelow_mono (le_self_add : n ≤ n +1)) IH | case succ.refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap 0 n ∈ Subgroup.closure (adjSwapsBelow (n + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Subgroup.subset_closure ⟨n, by simp, rfl⟩ | case succ.refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap n (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ swap n (n + 1) ∈ Subgroup.closure (adjSwapsBelow (n + 1))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | simp | α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ n ∈ Set.Iio (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ n ∈ Set.Iio (n + 1)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.adjSwapsBelow_aux | [399, 1] | [410, 37] | exact Nat.ne_of_beq_eq_false rfl | case succ.hxz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ 0 ≠ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.hxz
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
IH : swap 0 n ∈ Subgroup.closure (adjSwapsBelow n)
⊢ 0 ≠ n + 1
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | refine le_antisymm ((Subgroup.closure_le _).2 ?_) ?_ | α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ Subgroup.closure (adjSwapsBelow n) = restrict (Set.Iic n) | case refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ adjSwapsBelow n ⊆ ↑(restrict (Set.Iic n))
case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ restrict (Set.Iic n) ≤ Subgroup.closure (adjSwapsBelow n) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ Subgroup.closure (adjSwapsBelow n) = restrict (Set.Iic n)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | rw [← cl_swapsAt_eq' (x := 0) (by simp)] | case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ restrict (Set.Iic n) ≤ Subgroup.closure (adjSwapsBelow n) | case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ Subgroup.closure (swapsAt 0 (Set.Iic n)) ≤ Subgroup.closure (adjSwapsBelow n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ restrict (Set.Iic n) ≤ Subgroup.closure (adjSwapsBelow n)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | apply (Subgroup.closure_le _).2 | case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ Subgroup.closure (swapsAt 0 (Set.Iic n)) ≤ Subgroup.closure (adjSwapsBelow n) | case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ swapsAt 0 (Set.Iic n) ⊆ ↑(Subgroup.closure (adjSwapsBelow n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ Subgroup.closure (swapsAt 0 (Set.Iic n)) ≤ Subgroup.closure (adjSwapsBelow n)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | rintro _ ⟨i, ⟨hi : i ≤ n, -⟩, rfl⟩ | case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ swapsAt 0 (Set.Iic n) ⊆ ↑(Subgroup.closure (adjSwapsBelow n)) | case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ (fun x => swap 0 x) i ∈ ↑(Subgroup.closure (adjSwapsBelow n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ swapsAt 0 (Set.Iic n) ⊆ ↑(Subgroup.closure (adjSwapsBelow n))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | simp only [SetLike.mem_coe] | case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ (fun x => swap 0 x) i ∈ ↑(Subgroup.closure (adjSwapsBelow n)) | case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ swap 0 i ∈ Subgroup.closure (adjSwapsBelow n) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ (fun x => swap 0 x) i ∈ ↑(Subgroup.closure (adjSwapsBelow n))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | exact Subgroup.closure_mono (adjSwapsBelow_mono hi) <| adjSwapsBelow_aux i | case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ swap 0 i ∈ Subgroup.closure (adjSwapsBelow n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.intro.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ≤ n
⊢ swap 0 i ∈ Subgroup.closure (adjSwapsBelow n)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | rintro x ⟨i, hi, rfl⟩ | case refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ adjSwapsBelow n ⊆ ↑(restrict (Set.Iic n)) | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ (fun i => swap i (i + 1)) i ∈ ↑(restrict (Set.Iic n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ adjSwapsBelow n ⊆ ↑(restrict (Set.Iic n))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | simp only [SetLike.mem_coe, mem_restrict_iff, swap_support, self_eq_add_right, ite_false,
coe_insert, coe_singleton, Set.insert_subset_iff, Set.mem_Iic, Set.singleton_subset_iff] | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ (fun i => swap i (i + 1)) i ∈ ↑(restrict (Set.Iic n)) | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ i ≤ n ∧ i + 1 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ (fun i => swap i (i + 1)) i ∈ ↑(restrict (Set.Iic n))
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | rw [Set.mem_Iio, ← Nat.add_one_le_iff] at hi | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ i ≤ n ∧ i + 1 ≤ n | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i + 1 ≤ n
⊢ i ≤ n ∧ i + 1 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i ∈ Set.Iio n
⊢ i ≤ n ∧ i + 1 ≤ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | constructor <;> linarith | case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i + 1 ≤ n
⊢ i ≤ n ∧ i + 1 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro.intro
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n i : ℕ
hi : i + 1 ≤ n
⊢ i ≤ n ∧ i + 1 ≤ n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwapsBelow_eq | [412, 1] | [424, 77] | simp | α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ 0 ∈ Set.Iic n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
n : ℕ
⊢ 0 ∈ Set.Iic n
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | ext f | α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ Subgroup.closure adjSwaps = ⊤ | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps ↔ f ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ g : Finperm α
f : Finperm ℕ
⊢ Subgroup.closure adjSwaps = ⊤
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | simp only [Subgroup.mem_top, iff_true] | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps ↔ f ∈ ⊤ | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps ↔ f ∈ ⊤
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | have hss : (f.support : Set ℕ) ⊆ Set.Iic f.ub | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps | case hss
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ ↑f.support ⊆ Set.Iic (ub f)
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
⊢ f ∈ Subgroup.closure adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ f ∈ Subgroup.closure adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | have hf := restrict_mono hss <| mem_restrict_support f | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
⊢ f ∈ Subgroup.closure adjSwaps | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ restrict (Set.Iic (ub f))
⊢ f ∈ Subgroup.closure adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
⊢ f ∈ Subgroup.closure adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | rw [← closure_adjSwapsBelow_eq] at hf | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ restrict (Set.Iic (ub f))
⊢ f ∈ Subgroup.closure adjSwaps | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ f ∈ Subgroup.closure adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ restrict (Set.Iic (ub f))
⊢ f ∈ Subgroup.closure adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | refine (Subgroup.closure_mono ?_) hf | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ f ∈ Subgroup.closure adjSwaps | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ adjSwapsBelow (ub f) ⊆ adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ f ∈ Subgroup.closure adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | rintro _ ⟨i, -, rfl⟩ | case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ adjSwapsBelow (ub f) ⊆ adjSwaps | case h.intro.intro
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
i : ℕ
⊢ (fun i => swap i (i + 1)) i ∈ adjSwaps | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
⊢ adjSwapsBelow (ub f) ⊆ adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | exact ⟨i, rfl⟩ | case h.intro.intro
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
i : ℕ
⊢ (fun i => swap i (i + 1)) i ∈ adjSwaps | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
hss : ↑f.support ⊆ Set.Iic (ub f)
hf : f ∈ Subgroup.closure (adjSwapsBelow (ub f))
i : ℕ
⊢ (fun i => swap i (i + 1)) i ∈ adjSwaps
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Finperm.lean | Finperm.closure_adjSwaps_eq | [428, 1] | [437, 17] | exact fun x hx ↦ (f.lt_ub hx).le | case hss
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ ↑f.support ⊆ Set.Iic (ub f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hss
α : Type u_1
x y : α
f✝¹ g : Finperm α
f✝ f : Finperm ℕ
⊢ ↑f.support ⊆ Set.Iic (ub f)
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.not_mem_support' | [25, 9] | [26, 18] | simp [support'] | α : Type u_1
x y : α
f✝ f : Perm α
⊢ x ∉ support' f ↔ f x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
⊢ x ∉ support' f ↔ f x = x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.mem_support' | [28, 9] | [29, 18] | simp [support'] | α : Type u_1
x y : α
f✝ f : Perm α
⊢ x ∈ support' f ↔ f x ≠ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x y : α
f✝ f : Perm α
⊢ x ∈ support' f ↔ f x ≠ x
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_one | [31, 9] | [32, 12] | ext | α✝ : Type u_1
x y : α✝
f : Perm α✝
α : Type u_2
⊢ support' 1 = ∅ | case h
α✝ : Type u_1
x y : α✝
f : Perm α✝
α : Type u_2
x✝ : α
⊢ x✝ ∈ support' 1 ↔ x✝ ∈ ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x y : α✝
f : Perm α✝
α : Type u_2
⊢ support' 1 = ∅
TACTIC:
|
https://github.com/mguaypaq/lean-bruhat.git | 1666a1bee2b520d5ba8a662310b4bd257fcf7ac2 | Bruhat/Perm.lean | Equiv.Perm.support'_one | [31, 9] | [32, 12] | simp | case h
α✝ : Type u_1
x y : α✝
f : Perm α✝
α : Type u_2
x✝ : α
⊢ x✝ ∈ support' 1 ↔ x✝ ∈ ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type u_1
x y : α✝
f : Perm α✝
α : Type u_2
x✝ : α
⊢ x✝ ∈ support' 1 ↔ x✝ ∈ ∅
TACTIC:
|
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