internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/mguaypaq/lean-bruhat.git	1666a1bee2b520d5ba8a662310b4bd257fcf7ac2	Bruhat/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}
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}
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}
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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
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)
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
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
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
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
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
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
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
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
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
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
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
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)
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
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)
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)
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))
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))
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))
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))
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
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)
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)
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)
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
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)
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
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))
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
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
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))
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))
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
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
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)
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
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))
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
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
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
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
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
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)
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)
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))
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))
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)
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
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))
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
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
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
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
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 ∈ ⊤
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
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
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
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
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
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
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
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
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
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
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✝ ∈ ∅
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

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}

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}

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}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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

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)

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

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

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

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

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

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

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

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

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

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

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

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)

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

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)

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)

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))

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))

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))

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))

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

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)

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)

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)

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

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)

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

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))

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

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

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))

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))

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

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

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)

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

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))

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

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

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

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

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

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)

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)

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))

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))

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)

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

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))

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

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

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

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

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 ∈ ⊤

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

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

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

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

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

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

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

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

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

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

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✝ ∈ ∅

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