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/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | refine' Subgroup.mem_top g1 | case mpr.comm.intro.intro.a.h₁
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g1 ∈ ⊤
case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤ | case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a.h₁
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g1 ∈ ⊤
case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.Subgroup.quotient_isCommutative_iff_commutator_le | [283, 1] | [304, 32] | refine' Subgroup.mem_top g2 | case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.comm.intro.intro.a.h₂
α : Type ?u.69108
inst✝² : DecidableEq α
inst✝¹ : Fintype α
G : Type u_1
inst✝ : Group G
H : Subgroup G
nH : Subgroup.Normal H
h : commutator G ≤ H
g1 g2 : G
⊢ g2 ∈ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [eq_bot_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
⊢ Subgroup.center ↥(alternatingGroup α) = ⊥ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
⊢ Subgroup.center ↥(alternatingGroup α) ≤ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
⊢ Subgroup.center ↥(alternatingGroup α) = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rintro ⟨g, hg⟩ hg' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
⊢ Subgroup.center ↥(alternatingGroup α) ≤ ⊥ | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
⊢ Subgroup.center ↥(alternatingGroup α) ≤ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp only [Subgroup.mem_bot] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } ∈ ⊥ | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } ∈ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp only [← Subtype.coe_inj, Subgroup.coe_mk, Subgroup.coe_one] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } = 1 | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ { val := g, property := hg } = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [← Equiv.Perm.support_eq_empty_iff] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ g = 1 | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ Equiv.Perm.support g = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ g = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [Finset.eq_empty_iff_forall_not_mem] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ Equiv.Perm.support g = ∅ | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ ∀ (x : α), x ∉ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ Equiv.Perm.support g = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | intro a ha | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ ∀ (x : α), x ∉ Equiv.Perm.support g | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
⊢ ∀ (x : α), x ∉ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | let b := g a | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
⊢ False | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | have hab : b ≠ a := by simp only [b]; rw [← Equiv.Perm.mem_support]; exact ha | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ False | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | have : ({a, b} : Finset α)ᶜ.Nonempty :=
by
rw [← Finset.card_compl_lt_iff_nonempty, compl_compl, Finset.card_pair hab.symm]
refine' lt_of_lt_of_le (by norm_num) hα4 | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ False | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
this : {a, b}ᶜ.Nonempty
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | obtain ⟨c, hc⟩ := this | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
this : {a, b}ᶜ.Nonempty
⊢ False | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : c ∈ {a, b}ᶜ
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
this : {a, b}ᶜ.Nonempty
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp only [Finset.compl_insert, Finset.mem_erase, Ne.def, Finset.mem_compl,
Finset.mem_singleton] at hc | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : c ∈ {a, b}ᶜ
⊢ False | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : c ∈ {a, b}ᶜ
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | have h2trans : MulAction.IsMultiplyPretransitive (alternatingGroup α) α 2 := by
refine' MulAction.isMultiplyPretransitive_of_higher (alternatingGroup α) α
(MulAction.IsMultiplyPretransitive.alternatingGroup_of_sub_two α) _ _
exact Nat.le_sub_of_add_le hα4
simp | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ False | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : MulAction.IsMultiplyPretransitive (↥(alternatingGroup α)) α 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [MulAction.is_two_pretransitive_iff] at h2trans | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : MulAction.IsMultiplyPretransitive (↥(alternatingGroup α)) α 2
⊢ False | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : MulAction.IsMultiplyPretransitive (↥(alternatingGroup α)) α 2
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | obtain ⟨k, hk, hk'⟩ := h2trans a b a c hab.symm (Ne.symm hc.left) | case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
⊢ False | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | suffices k • (⟨g, hg⟩ : alternatingGroup α) • a ≠ c by
apply this; rw [← hk']; rfl | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ False | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a ≠ c | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | suffices k • (⟨g, hg⟩ : alternatingGroup α) • a = (⟨g, hg⟩ : alternatingGroup α) • k • a by
rw [this, hk]; exact Ne.symm hc.right | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a ≠ c | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a ≠ c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [Subgroup.mem_center_iff] at hg' | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : ∀ (g_1 : ↥(alternatingGroup α)), g_1 * { val := g, property := hg } = { val := g, property := hg } * g_1
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | specialize hg' k | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : ∀ (g_1 : ↥(alternatingGroup α)), g_1 * { val := g, property := hg } = { val := g, property := hg } * g_1
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
hg' : k * { val := g, property := hg } = { val := g, property := hg } * k
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : ∀ (g_1 : ↥(alternatingGroup α)), g_1 * { val := g, property := hg } = { val := g, property := hg } * g_1
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp only [smul_smul, hg'] | case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
hg' : k * { val := g, property := hg } = { val := g, property := hg } * k
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
hg' : k * { val := g, property := hg } = { val := g, property := hg } * k
⊢ k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp only [b] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ b ≠ a | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ g a ≠ a | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ b ≠ a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [← Equiv.Perm.mem_support] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ g a ≠ a | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ a ∈ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ g a ≠ a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | exact ha | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ a ∈ Equiv.Perm.support g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
⊢ a ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [← Finset.card_compl_lt_iff_nonempty, compl_compl, Finset.card_pair hab.symm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ {a, b}ᶜ.Nonempty | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ 2 < Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ {a, b}ᶜ.Nonempty
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | refine' lt_of_lt_of_le (by norm_num) hα4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ 2 < Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ 2 < Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ 2 < 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
⊢ 2 < 4
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | refine' MulAction.isMultiplyPretransitive_of_higher (alternatingGroup α) α
(MulAction.IsMultiplyPretransitive.alternatingGroup_of_sub_two α) _ _ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ MulAction.IsMultiplyPretransitive (↥(alternatingGroup α)) α 2 | case refine'_1
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ 2 ≤ Fintype.card α - 2
case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ MulAction.IsMultiplyPretransitive (↥(alternatingGroup α)) α 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | exact Nat.le_sub_of_add_le hα4 | case refine'_1
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ 2 ≤ Fintype.card α - 2
case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_1
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ 2 ≤ Fintype.card α - 2
case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | simp | case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine'_2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
⊢ ↑(Fintype.card α - 2) ≤ PartENat.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | apply this | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ False | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [← hk'] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = c | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = k • b | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = k • b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a ≠ c
⊢ k • { val := g, property := hg } • a = k • b
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | rw [this, hk] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
⊢ k • { val := g, property := hg } • a ≠ c | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
⊢ { val := g, property := hg } • a ≠ c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
⊢ k • { val := g, property := hg } • a ≠ c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.center_bot | [307, 1] | [337, 29] | exact Ne.symm hc.right | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
⊢ { val := g, property := hg } • a ≠ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : 4 ≤ Fintype.card α
g : Equiv.Perm α
hg : g ∈ alternatingGroup α
hg' : { val := g, property := hg } ∈ Subgroup.center ↥(alternatingGroup α)
a : α
ha : a ∈ Equiv.Perm.support g
b : α := g a
hab : b ≠ a
c : α
hc : ¬c = a ∧ ¬c = b
h2trans : ∀ (a b c d : α), a ≠ b → c ≠ d → ∃ m, m • a = c ∧ m • b = d
k : ↥(alternatingGroup α)
hk : k • a = a
hk' : k • b = c
this : k • { val := g, property := hg } • a = { val := g, property := hg } • k • a
⊢ { val := g, property := hg } • a ≠ c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | haveI : (V4 α).Normal := V4_is_normal α hα4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ V4 α = commutator ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ V4 α = commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ V4 α = commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | have comm_ne_bot : commutator (alternatingGroup α) ≠ ⊥ := by
have : Nontrivial (Subgroup (alternatingGroup α)) := by
rw [Subgroup.nontrivial_iff]
rw [← Fintype.one_lt_card_iff_nontrivial]
rw [A4_card α hα4]
norm_num
rw [Ne.def, commutator, Subgroup.commutator_eq_bot_iff_le_centralizer,
← eq_top_iff, Subgroup.coe_top, Subgroup.centralizer_univ,
alternatingGroup.center_bot _]
exact bot_ne_top
rw [hα4] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ V4 α = commutator ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
⊢ V4 α = commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ V4 α = commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | obtain ⟨k, hk, hk'⟩ := Or.resolve_left (Subgroup.bot_or_exists_ne_one _) comm_ne_bot | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
⊢ V4 α = commutator ↥(alternatingGroup α) | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
⊢ V4 α = commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
⊢ V4 α = commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | have hk2 := comm_le hk | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ V4 α
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← Subgroup.mem_carrier, V4_carrier_eq α hα4] at hk2 | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ V4 α
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ V4 α
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | cases' hk2 with hk2 hk2 | case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
case intro.intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : k ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← Subgroup.quotient_isCommutative_iff_commutator_le] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ commutator ↥(alternatingGroup α) ≤ V4 α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Std.Commutative fun x x_1 => x * x_1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ commutator ↥(alternatingGroup α) ≤ V4 α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | apply isCommutative_of_prime_order (hp := Nat.fact_prime_three) _ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Std.Commutative fun x x_1 => x * x_1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Std.Commutative fun x x_1 => x * x_1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | apply mul_left_injective₀ _ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) = 3 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ (fun a => a * ?m.92416) (Fintype.card (↥(alternatingGroup α) ⧸ V4 α)) = (fun a => a * ?m.92416) 3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) = 3
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | dsimp | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ (fun a => a * ?m.92416) (Fintype.card (↥(alternatingGroup α) ⧸ V4 α)) = (fun a => a * ?m.92416) 3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) * ?m.92416 = 3 * ?m.92416
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ (fun a => a * ?m.92416) (Fintype.card (↥(alternatingGroup α) ⧸ V4 α)) = (fun a => a * ?m.92416) 3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← Subgroup.card_eq_card_quotient_mul_card_subgroup, V4_card α hα4, A4_card α hα4] | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) * ?m.92416 = 3 * ?m.92416
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card ↥(V4 α) ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card (↥(alternatingGroup α) ⧸ V4 α) * ?m.92416 = 3 * ?m.92416
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ℕ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ ?m.92416 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card ↥(V4 α) ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
⊢ Fintype.card ↥(V4 α) ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | have : Nontrivial (Subgroup (alternatingGroup α)) := by
rw [Subgroup.nontrivial_iff]
rw [← Fintype.one_lt_card_iff_nontrivial]
rw [A4_card α hα4]
norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ commutator ↥(alternatingGroup α) ≠ ⊥ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ commutator ↥(alternatingGroup α) ≠ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ commutator ↥(alternatingGroup α) ≠ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [Ne.def, commutator, Subgroup.commutator_eq_bot_iff_le_centralizer,
← eq_top_iff, Subgroup.coe_top, Subgroup.centralizer_univ,
alternatingGroup.center_bot _] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ commutator ↥(alternatingGroup α) ≠ ⊥ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ ¬⊥ = ⊤
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ commutator ↥(alternatingGroup α) ≠ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | exact bot_ne_top | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ ¬⊥ = ⊤
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ ¬⊥ = ⊤
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [hα4] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
this : Nontrivial (Subgroup ↥(alternatingGroup α))
⊢ 4 ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [Subgroup.nontrivial_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ Nontrivial (Subgroup ↥(alternatingGroup α)) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ Nontrivial ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ Nontrivial (Subgroup ↥(alternatingGroup α))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← Fintype.one_lt_card_iff_nontrivial] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ Nontrivial ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < Fintype.card ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ Nontrivial ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [A4_card α hα4] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < Fintype.card ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < 12 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < Fintype.card ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < 12 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
⊢ 1 < 12
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | refine' le_antisymm _ comm_le | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ V4 α = commutator ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ V4 α ≤ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ V4 α = commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | intro g hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ V4 α ≤ commutator ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ V4 α
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ V4 α ≤ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← Subgroup.mem_carrier, V4_carrier_eq α hα4] at hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ V4 α
⊢ g ∈ commutator ↥(alternatingGroup α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ V4 α
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | cases' hg with hg hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ commutator ↥(alternatingGroup α) | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = 0
⊢ g ∈ commutator ↥(alternatingGroup α)
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g ∈ {g | Equiv.Perm.cycleType ↑g = 0 ∨ Equiv.Perm.cycleType ↑g = {2, 2}}
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [Equiv.Perm.cycleType_eq_zero, OneMemClass.coe_eq_one] at hg | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = 0
⊢ g ∈ commutator ↥(alternatingGroup α) | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = 0
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [hg] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ g ∈ commutator ↥(alternatingGroup α) | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ 1 ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | exact Subgroup.one_mem _ | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ 1 ∈ commutator ↥(alternatingGroup α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : g = 1
⊢ 1 ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← hg, ← Equiv.Perm.isConj_iff_cycleType_eq, isConj_iff] at hk22 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α) | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hk22 : ∃ c, c * ↑k * c⁻¹ = ↑g
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk22 : Equiv.Perm.cycleType ↑k = {2, 2}
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | obtain ⟨c, hc⟩ := hk22 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hk22 : ∃ c, c * ↑k * c⁻¹ = ↑g
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α) | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : c * ↑k * c⁻¹ = ↑g
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hk22 : ∃ c, c * ↑k * c⁻¹ = ↑g
hg : Equiv.Perm.cycleType ↑g = {2, 2}
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← MulAut.conjNormal_apply, Subtype.coe_inj] at hc | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : c * ↑k * c⁻¹ = ↑g
⊢ g ∈ commutator ↥(alternatingGroup α) | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ g ∈ commutator ↥(alternatingGroup α) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : c * ↑k * c⁻¹ = ↑g
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | simp only [commutator, ← hc] | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ g ∈ commutator ↥(alternatingGroup α) | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ g ∈ commutator ↥(alternatingGroup α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | let fc : MulAut (alternatingGroup α) := MulAut.conjNormal c | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆ | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | suffices (⊤ : Subgroup (alternatingGroup α)) =
Subgroup.map fc.toMonoidHom (⊤ : Subgroup (alternatingGroup α)) by
rw [this, ← Subgroup.map_commutator]
refine' Subgroup.mem_map_of_mem _ hk | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆ | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | apply symm | case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤ | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Subgroup.map (MulEquiv.toMonoidHom fc) ⊤ = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [← MonoidHom.range_eq_map] | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Subgroup.map (MulEquiv.toMonoidHom fc) ⊤ = ⊤ | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ MonoidHom.range (MulEquiv.toMonoidHom fc) = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Subgroup.map (MulEquiv.toMonoidHom fc) ⊤ = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [MonoidHom.range_top_iff_surjective] | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ MonoidHom.range (MulEquiv.toMonoidHom fc) = ⊤ | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom fc) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ MonoidHom.range (MulEquiv.toMonoidHom fc) = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | exact MulEquiv.surjective _ | case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom fc) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
⊢ Function.Surjective ⇑(MulEquiv.toMonoidHom fc)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [this, ← Subgroup.map_commutator] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
this : ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
this : ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
⊢ (MulAut.conjNormal c) k ∈ Subgroup.map (MulEquiv.toMonoidHom fc) ⁅⊤, ⊤⁆ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
this : ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
⊢ (MulAut.conjNormal c) k ∈ ⁅⊤, ⊤⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | refine' Subgroup.mem_map_of_mem _ hk | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
this : ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
⊢ (MulAut.conjNormal c) k ∈ Subgroup.map (MulEquiv.toMonoidHom fc) ⁅⊤, ⊤⁆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this✝ : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
g : ↥(alternatingGroup α)
hg : Equiv.Perm.cycleType ↑g = {2, 2}
c : Equiv.Perm α
hc : (MulAut.conjNormal c) k = g
fc : MulAut ↥(alternatingGroup α) := MulAut.conjNormal c
this : ⊤ = Subgroup.map (MulEquiv.toMonoidHom fc) ⊤
⊢ (MulAut.conjNormal c) k ∈ Subgroup.map (MulEquiv.toMonoidHom fc) ⁅⊤, ⊤⁆
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | exfalso | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | apply hk' | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ False | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | rw [Equiv.Perm.cycleType_eq_zero] at hk2 | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1 | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : ↑k = 1
⊢ k = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = 0
⊢ k = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | simp only [← Subtype.coe_inj, hk2, Subgroup.coe_one] | case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : ↑k = 1
⊢ k = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : ↑k = 1
⊢ k = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.V4_eq_commutator | [340, 1] | [390, 14] | exact hk2 | case intro.intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Subgroup.Normal (V4 α)
comm_le : commutator ↥(alternatingGroup α) ≤ V4 α
comm_ne_bot : commutator ↥(alternatingGroup α) ≠ ⊥
k : ↥(alternatingGroup α)
hk : k ∈ commutator ↥(alternatingGroup α)
hk' : k ≠ 1
hk2 : Equiv.Perm.cycleType ↑k = {2, 2}
⊢ Equiv.Perm.cycleType ↑k = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | constructor | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → IsBlock G B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | intro hB g g' | case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B → ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | by_cases h : g • B = g' • B | case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case pos
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
case neg
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | refine' Or.intro_left _ h | case pos
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | apply Or.intro_right | case neg
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B) | case neg.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ Disjoint (g • B) (g' • B) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ g • B = g' • B ∨ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | exact hB (Set.mem_range_self g) (Set.mem_range_self g') h | case neg.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ Disjoint (g • B) (g' • B) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : IsBlock G B
g g' : G
h : ¬g • B = g' • B
⊢ Disjoint (g • B) (g' • B)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | intro hB | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → IsBlock G B | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ IsBlock G B | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) → IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | unfold IsBlock | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ IsBlock G B | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ Set.PairwiseDisjoint (Set.range fun g => g • B) id | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ IsBlock G B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | intro C hC C' hC' | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ Set.PairwiseDisjoint (Set.range fun g => g • B) id | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C : Set X
hC : C ∈ Set.range fun g => g • B
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
⊢ C ≠ C' → (Disjoint on id) C C' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
⊢ Set.PairwiseDisjoint (Set.range fun g => g • B) id
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | obtain ⟨g, rfl⟩ := hC | case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C : Set X
hC : C ∈ Set.range fun g => g • B
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
⊢ C ≠ C' → (Disjoint on id) C C' | case mpr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
g : G
⊢ (fun g => g • B) g ≠ C' → (Disjoint on id) ((fun g => g • B) g) C' | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C : Set X
hC : C ∈ Set.range fun g => g • B
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
⊢ C ≠ C' → (Disjoint on id) C C'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | obtain ⟨g', rfl⟩ := hC' | case mpr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
g : G
⊢ (fun g => g • B) g ≠ C' → (Disjoint on id) ((fun g => g • B) g) C' | case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
⊢ (fun g => g • B) g ≠ (fun g => g • B) g' → (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
C' : Set X
hC' : C' ∈ Set.range fun g => g • B
g : G
⊢ (fun g => g • B) g ≠ C' → (Disjoint on id) ((fun g => g • B) g) C'
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | intro hgg' | case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
⊢ (fun g => g • B) g ≠ (fun g => g • B) g' → (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
⊢ (fun g => g • B) g ≠ (fun g => g • B) g' → (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | cases hB g g' with
| inl h => exfalso; exact hgg' h
| inr h => exact h | case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | exfalso | case mpr.intro.intro.inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : g • B = g' • B
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | case mpr.intro.intro.inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : g • B = g' • B
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : g • B = g' • B
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | exact hgg' h | case mpr.intro.intro.inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : g • B = g' • B
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.inl
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : g • B = g' • B
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.def | [65, 1] | [82, 23] | exact h | case mpr.intro.intro.inr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : Disjoint (g • B) (g' • B)
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.inr
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
hB : ∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)
g g' : G
hgg' : (fun g => g • B) g ≠ (fun g => g • B) g'
h : Disjoint (g • B) (g' • B)
⊢ (Disjoint on id) ((fun g => g • B) g) ((fun g => g • B) g')
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | rw [IsBlock.def] | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ IsBlock G B ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | apply forall_congr' | G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ ∀ (a : G), (∀ (g' : G), a • B = g' • B ∨ Disjoint (a • B) (g' • B)) ↔ ∀ (g' : G), a • B ∩ g' • B ≠ ∅ → a • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ (∀ (g g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | intro g | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ ∀ (a : G), (∀ (g' : G), a • B = g' • B ∨ Disjoint (a • B) (g' • B)) ↔ ∀ (g' : G), a • B ∩ g' • B ≠ ∅ → a • B = g' • B | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ (∀ (g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
⊢ ∀ (a : G), (∀ (g' : G), a • B = g' • B ∨ Disjoint (a • B) (g' • B)) ↔ ∀ (g' : G), a • B ∩ g' • B ≠ ∅ → a • B = g' • B
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/Blocks.lean | MulAction.IsBlock.mk_notempty | [86, 1] | [93, 29] | apply forall_congr' | case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ (∀ (g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B | case h.h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ ∀ (a : G), g • B = a • B ∨ Disjoint (g • B) (a • B) ↔ g • B ∩ a • B ≠ ∅ → g • B = a • B | Please generate a tactic in lean4 to solve the state.
STATE:
case h
G : Type u_2
X : Type u_1
inst✝ : SMul G X
B : Set X
g : G
⊢ (∀ (g' : G), g • B = g' • B ∨ Disjoint (g • B) (g' • B)) ↔ ∀ (g' : G), g • B ∩ g' • B ≠ ∅ → g • B = g' • B
TACTIC:
|
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