url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.IsPretransitive.of_partition | [77, 1] | [102, 45] | rw [MulAction.mul_smul, hgab, hk] | case h
α : Type u_1
inst✝² : DecidableEq α
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
hs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
hG : stabilizer G s ≠ ⊤
a b : α
g : G
ha : a ∈ s
hb : b ∈ sᶜ
hgab : g • a = b
x : α
hx : x ∈ sᶜ
k : G
hk : k • b = x
⊢ (k * g) • a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : DecidableEq α
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
hs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b
hs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b
hG : stabilizer G s ≠ ⊤
a b : α
g : G
ha : a ∈ s
hb : b ∈ sᶜ
hgab : g • a = b
x : α
hx : x ∈ sᶜ
k : G
hk : k • b = x
⊢ (k * g) • a = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | suffices Equiv.swap a b • s ⊆ s by
rw [mem_stabilizer_iff]
apply Set.Subset.antisymm
exact this
exact Set.subset_set_smul_iff.mpr this | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b ∈ stabilizer (Perm α) s | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b • s ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | rintro _ ⟨x, hx, rfl⟩ | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b • s ⊆ s | case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (fun x => swap a b • x) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
⊢ swap a b • s ⊆ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | simp only [Equiv.Perm.smul_def] | case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (fun x => swap a b • x) x ∈ s | case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (fun x => swap a b • x) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | cases' em (x = a) with hxa hxa' | case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (swap a b) x ∈ s | case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ (swap a b) x ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | rw [hxa, Equiv.swap_apply_left] | case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ (swap a b) x ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ b ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ (swap a b) x ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | exact hb | case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ b ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa : x = a
⊢ b ∈ s
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | cases' em (x = b) with hxb hxb' | case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s | case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ (swap a b) x ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | rw [hxb, Equiv.swap_apply_right] | case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ (swap a b) x ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ a ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ (swap a b) x ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | exact ha | case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ a ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.inl
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb : x = b
⊢ a ∈ s
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | rw [Equiv.swap_apply_of_ne_of_ne hxa' hxb'] | case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s | case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ (swap a b) x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | exact hx | case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ x ∈ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr.inr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
x : α
hx : x ∈ s
hxa' : ¬x = a
hxb' : ¬x = b
⊢ x ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | rw [mem_stabilizer_iff] | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b ∈ stabilizer (Perm α) s | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s = s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | apply Set.Subset.antisymm | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s = s | case h₁
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s ⊆ s
case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s = s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | exact this | case h₁
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s ⊆ s
case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s | case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ swap a b • s ⊆ s
case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_mem_stabilizer | [105, 1] | [118, 56] | exact Set.subset_set_smul_iff.mpr this | case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.11084
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
s : Set α
ha : a ∈ s
hb : b ∈ s
this : swap a b • s ⊆ s
⊢ s ⊆ swap a b • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | intro h1 | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
hf : IsSwap f
⊢ f ≠ 1 | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
hf : IsSwap f
h1 : f = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
hf : IsSwap f
⊢ f ≠ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | obtain ⟨x, y, hxy, h⟩ := hf | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
hf : IsSwap f
h1 : f = 1
⊢ False | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : f = swap x y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
hf : IsSwap f
h1 : f = 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | rw [h1] at h | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : f = swap x y
⊢ False | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : f = swap x y
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | apply hxy | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ False | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | rw [← Equiv.swap_apply_left x y] | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = y | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = (swap x y) x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | rw [← h] | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = (swap x y) x | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = (swap x y) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.ne_one_of_isSwap | [121, 1] | [126, 41] | simp only [Equiv.Perm.coe_one, id.def] | case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.14379
inst✝¹ : Group G
inst✝ : MulAction G α
f : Perm α
h1 : f = 1
x y : α
hxy : x ≠ y
h : 1 = swap x y
⊢ x = 1 x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | constructor | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ IsSwap (swap a b) ↔ a ≠ b | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ IsSwap (swap a b) → a ≠ b
case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ a ≠ b → IsSwap (swap a b) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ IsSwap (swap a b) ↔ a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | intro h hab | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ IsSwap (swap a b) → a ≠ b | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ IsSwap (swap a b) → a ≠ b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | suffices Equiv.swap a b ≠ 1 by
apply this
rw [← hab, Equiv.swap_self]
ext x; simp only [Equiv.coe_refl, Equiv.Perm.coe_one] | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ False | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ swap a b ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | exact ne_one_of_isSwap h | case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ swap a b ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
⊢ swap a b ≠ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | apply this | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ False | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ swap a b = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | rw [← hab, Equiv.swap_self] | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ swap a b = 1 | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ Equiv.refl α = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ swap a b = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | ext x | α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ Equiv.refl α = 1 | case H
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
x : α
⊢ (Equiv.refl α) x = 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
⊢ Equiv.refl α = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | simp only [Equiv.coe_refl, Equiv.Perm.coe_one] | case H
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
x : α
⊢ (Equiv.refl α) x = 1 x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : IsSwap (swap a b)
hab : a = b
this : swap a b ≠ 1
x : α
⊢ (Equiv.refl α) x = 1 x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | intro h | case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ a ≠ b → IsSwap (swap a b) | case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ IsSwap (swap a b) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
⊢ a ≠ b → IsSwap (swap a b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | use a | case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ IsSwap (swap a b) | case h
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ ∃ y, a ≠ y ∧ swap a b = swap a y | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ IsSwap (swap a b)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.swap_isSwap_iff | [129, 1] | [138, 27] | use b | case h
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ ∃ y, a ≠ y ∧ swap a b = swap a y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝² : DecidableEq α
G : Type ?u.15398
inst✝¹ : Group G
inst✝ : MulAction G α
a b : α
h : a ≠ b
⊢ ∃ y, a ≠ y ∧ swap a b = swap a y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | intro a ha b hb | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
⊢ ∀ a ∈ t, ∀ b ∈ t, ∃ g, g • a = b | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ ∃ g, g • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
⊢ ∀ a ∈ t, ∀ b ∈ t, ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | use ⟨Equiv.swap a b, ?_⟩ | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ ∃ g, g • a = b | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ { val := swap a b, property := ?w } • a = b
case w
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ ∃ g, g • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | change Equiv.swap a b • a = b | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ { val := swap a b, property := ?w } • a = b | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ { val := swap a b, property := ?w } • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | simp only [Equiv.Perm.smul_def, Equiv.swap_apply_left] | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b • a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | apply le_of_lt hGt | case w
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ G | case w.a
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case w
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.moves_in | [168, 1] | [177, 36] | apply swap_mem_stabilizer ha hb | case w.a
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ stabilizer (Perm α) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case w.a
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.17243
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
G : Subgroup (Perm α)
t : Set α
hGt : stabilizer (Perm α) t < G
a : α
ha : a ∈ t
b : α
hb : b ∈ t
⊢ swap a b ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | obtain ⟨a, ha⟩ := hs | α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
hs : Set.Nonempty s
hsc : Set.Nonempty sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
hsc : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
hs : Set.Nonempty s
hsc : Set.Nonempty sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | obtain ⟨b, hb⟩ := hsc | case intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
hsc : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
hsc : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | intro h | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | rw [Set.mem_compl_iff] at hb | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | apply hb | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ False | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | have hg : Equiv.swap a b ∈ stabilizer (Equiv.Perm α) s := by rw [h]; apply Subgroup.mem_top | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ b ∈ s | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b ∈ stabilizer (Perm α) s
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | rw [mem_stabilizer_iff] at hg | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b ∈ stabilizer (Perm α) s
⊢ b ∈ s | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b ∈ stabilizer (Perm α) s
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | rw [← hg] | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ s | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ swap a b • s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | rw [Set.mem_smul_set] | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ swap a b • s | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ ∃ y ∈ s, swap a b • y = b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ b ∈ swap a b • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | use a | case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ ∃ y ∈ s, swap a b • y = b | case h
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ a ∈ s ∧ swap a b • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ ∃ y ∈ s, swap a b • y = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | use ha | case h
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ a ∈ s ∧ swap a b • a = b | case right
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ swap a b • a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ a ∈ s ∧ swap a b • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | apply Equiv.swap_apply_left | case right
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ swap a b • a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
hg : swap a b • s = s
⊢ swap a b • a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | rw [h] | α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ stabilizer (Perm α) s | α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ stabilizer (Perm α) s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_ne_top | [180, 1] | [190, 45] | apply Subgroup.mem_top | α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G : Type ?u.20454
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : Fintype α
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
⊢ swap a b ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_empty_eq_top | [193, 1] | [198, 33] | rw [eq_top_iff] | α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ stabilizer G ∅ = ⊤ | α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ stabilizer G ∅ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_empty_eq_top | [193, 1] | [198, 33] | intro g | α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G ∅ | α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_empty_eq_top | [193, 1] | [198, 33] | apply imp_intro | α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G ∅ | case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_empty_eq_top | [193, 1] | [198, 33] | rw [mem_stabilizer_iff] | case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G ∅ | case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • ∅ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_empty_eq_top | [193, 1] | [198, 33] | simp only [Set.smul_set_empty] | case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • ∅ = ∅ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.23342
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.23354
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • ∅ = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_univ_eq_top | [201, 1] | [206, 32] | rw [eq_top_iff] | α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ stabilizer G _root_.Set.univ = ⊤ | α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G _root_.Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ stabilizer G _root_.Set.univ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_univ_eq_top | [201, 1] | [206, 32] | intro g | α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G _root_.Set.univ | α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G _root_.Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
⊢ ⊤ ≤ stabilizer G _root_.Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_univ_eq_top | [201, 1] | [206, 32] | apply imp_intro | α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G _root_.Set.univ | case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G _root_.Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ ⊤ → g ∈ stabilizer G _root_.Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_univ_eq_top | [201, 1] | [206, 32] | rw [mem_stabilizer_iff] | case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G _root_.Set.univ | case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • _root_.Set.univ = _root_.Set.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g ∈ stabilizer G _root_.Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_univ_eq_top | [201, 1] | [206, 32] | simp only [Set.smul_set_univ] | case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • _root_.Set.univ = _root_.Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.25282
inst✝⁵ : DecidableEq α✝
G✝ : Type ?u.25294
inst✝⁴ : Group G✝
inst✝³ : MulAction G✝ α✝
inst✝² : Fintype α✝
G : Type u_1
inst✝¹ : Group G
α : Type u_2
inst✝ : MulAction G α
g : G
⊢ g • _root_.Set.univ = _root_.Set.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | obtain ⟨a : α, ha : a ∈ s⟩ := hs | α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
hs : Set.Nonempty s
hs' : Set.Nonempty sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
hs' : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
hs : Set.Nonempty s
hs' : Set.Nonempty sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | obtain ⟨b : α, hb : b ∈ sᶜ⟩ := hs' | case intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
hs' : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
hs' : Set.Nonempty sᶜ
a : α
ha : a ∈ s
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | intro h | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤ | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
⊢ stabilizer (Perm α) s ≠ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | let g := Equiv.swap a b | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | have h' : g ∈ ⊤ := Subgroup.mem_top g | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
⊢ False | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g ∈ ⊤
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | rw [← h, mem_stabilizer_iff] at h' | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g ∈ ⊤
⊢ False | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g ∈ ⊤
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | rw [Set.mem_compl_iff] at hb | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∈ sᶜ
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | apply hb | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | rw [← h'] | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ s | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ g • s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | use a | case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ g • s | case h
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ a ∈ s ∧ (fun x => g • x) a = b | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ b ∈ g • s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.stabilizer_nonempty_ne_top | [211, 1] | [224, 49] | exact And.intro ha (Equiv.swap_apply_left a b) | case h
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ a ∈ s ∧ (fun x => g • x) a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type ?u.27969
inst✝³ : DecidableEq α✝
G : Type ?u.27981
inst✝² : Group G
inst✝¹ : MulAction G α✝
inst✝ : Fintype α✝
α : Type u_1
s : Set α
a : α
ha : a ∈ s
b : α
hb : b ∉ s
h : stabilizer (Perm α) s = ⊤
g : Perm α := swap a b
h' : g • s = s
⊢ a ∈ s ∧ (fun x => g • x) a = b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | have : ∀ (t : Set α) (_ : 1 < t.ncard), ∃ (g : Equiv.Perm α),
g.IsSwap ∧ g ∈ stabilizer (Equiv.Perm α) t := by
intro t ht
rw [Set.one_lt_ncard_iff] at ht
obtain ⟨a, b, ha, hb, h⟩ := ht
simp only [Ne.def, Subtype.mk_eq_mk] at h
use Equiv.swap a b
constructor
rw [swap_isSwap_iff]; exact h
apply swap_mem_stabilizer ha hb | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∃ g, IsSwap g ∧ g ∈ G | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
⊢ ∃ g, IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | cases' lt_or_le 1 (s.ncard) with h1 h1' | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
⊢ ∃ g, IsSwap g ∧ g ∈ G | case inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
⊢ ∃ g, IsSwap g ∧ g ∈ G
case inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
⊢ ∃ g, IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | cases' lt_or_le 1 sᶜ.ncard with h1c h1c' | case inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
⊢ ∃ g, IsSwap g ∧ g ∈ G | case inr.inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G
case inr.inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ ∃ g, IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | cases subgroup_of_group_of_order_two (by
rw [Fintype.card_perm, ← Nat.card_eq_fintype_card, hα]
simp) G with
| inl h =>
exfalso; exact ne_bot_of_gt hG h
| inr h =>
rw [h]
rw [← stabilizer_univ_eq_top (Equiv.Perm α) α]
apply this
rw [Set.ncard_univ, hα]
norm_num | case inr.inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
hα : Nat.card α = 2
⊢ ∃ g, IsSwap g ∧ g ∈ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
hα : Nat.card α = 2
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | intro t ht | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : 1 < Set.ncard t
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
⊢ ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | rw [Set.one_lt_ncard_iff] at ht | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : 1 < Set.ncard t
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : ∃ a b, a ∈ t ∧ b ∈ t ∧ a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : 1 < Set.ncard t
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | obtain ⟨a, b, ha, hb, h⟩ := ht | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : ∃ a b, a ∈ t ∧ b ∈ t ∧ a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
ht : ∃ a b, a ∈ t ∧ b ∈ t ∧ a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | simp only [Ne.def, Subtype.mk_eq_mk] at h | case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : a ≠ b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | use Equiv.swap a b | case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b) ∧ swap a b ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | constructor | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b) ∧ swap a b ∈ stabilizer (Perm α) t | case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b)
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b) ∧ swap a b ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | rw [swap_isSwap_iff] | case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b)
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ a ≠ b
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ IsSwap (swap a b)
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | exact h | case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ a ≠ b
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ a ≠ b
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | apply swap_mem_stabilizer ha hb | case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
t : Set α
a b : α
ha : a ∈ t
hb : b ∈ t
h : ¬a = b
⊢ swap a b ∈ stabilizer (Perm α) t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | obtain ⟨g, hg, hg'⟩ := this s h1 | case inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
⊢ ∃ g, IsSwap g ∧ g ∈ G | case inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ ∃ g, IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | use g | case inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ ∃ g, IsSwap g ∧ g ∈ G | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | apply And.intro hg | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ IsSwap g ∧ g ∈ G | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | exact le_of_lt hG hg' | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1 : 1 < Set.ncard s
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | obtain ⟨g, hg, hg'⟩ := this sᶜ h1c | case inr.inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G | case inr.inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | use g | case inr.inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ IsSwap g ∧ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.intro.intro
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ ∃ g, IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | apply And.intro hg | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ IsSwap g ∧ g ∈ G | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ IsSwap g ∧ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | rw [stabilizer_compl] at hg' | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ g ∈ G | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) sᶜ
⊢ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | exact le_of_lt hG hg' | case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c : 1 < Set.ncard sᶜ
g : Perm α
hg : IsSwap g
hg' : g ∈ stabilizer (Perm α) s
⊢ g ∈ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | rw [← Set.ncard_add_ncard_compl s] | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Nat.card α = 2 | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s + Set.ncard sᶜ = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Nat.card α = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | convert Nat.one_add 1 | α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s + Set.ncard sᶜ = 2 | case h.e'_2.h.e'_5
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s = 1
case h.e'_2.h.e'_6
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard sᶜ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s + Set.ncard sᶜ = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/PermMaximal.lean | Equiv.Perm.has_swap_of_lt_stabilizer | [227, 1] | [274, 13] | apply le_antisymm | case h.e'_2.h.e'_5
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s = 1 | case h.e'_2.h.e'_5.a
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s ≤ 1
case h.e'_2.h.e'_5.a
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ 1 ≤ Set.ncard s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝³ : DecidableEq α
G✝ : Type ?u.29759
inst✝² : Group G✝
inst✝¹ : MulAction G✝ α
inst✝ : Fintype α
s : Set α
G : Subgroup (Perm α)
hG : stabilizer (Perm α) s < G
this : ∀ (t : Set α), 1 < Set.ncard t → ∃ g, IsSwap g ∧ g ∈ stabilizer (Perm α) t
h1' : Set.ncard s ≤ 1
h1c' : Set.ncard sᶜ ≤ 1
⊢ Set.ncard s = 1
TACTIC:
|
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