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/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply Nat.eq_of_lt_succ_of_not_lt | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) = 2 | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2 + 1
case a.h
α : Type ?... | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (Mul... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [Subgroup.index_ker f, Nat.card_eq_fintype_card, Nat.lt_succ_iff] | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2 + 1
case a.h
α : Type ?... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.range ⇑f) ≤ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
in... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ index (MonoidHom.ker ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply Nat.le_of_dvd two_pos | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.range ⇑f) ≤ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
in... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.range ⇑f) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
in... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.ra... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply dvd_trans f.range.card_subgroup_dvd_card | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.range ⇑f) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
in... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card (Equiv.Perm (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card ↑(Set.ra... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [Fintype.card_perm, ← Nat.card_eq_fintype_card] | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card (Equiv.Perm (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial (Nat.card (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype ... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Fintype.card (Equiv.P... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | unfold Subgroup.index at hH | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial (Nat.card (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype ... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial (Nat.card (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² :... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial (Nat.ca... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [hH] | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial (Nat.card (G ⧸ H)) ∣ 2
case a.h
α : Type ?u.28021
inst✝² :... | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial 2 ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factoria... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | norm_num | case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factorial 2 ∣ 2
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹... | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hmn
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : Nat.card (G ⧸ H) = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ Nat.factoria... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | intro h | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2 | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
⊢ ¬index (MonoidHom.ker (... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply Nat.not_succ_le_self 1 | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ False | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ Nat.succ 1 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | change 2 ≤ 1 | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ Nat.succ 1 ≤ 1 | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ 2 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [Nat.lt_succ_iff] at h | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) < 2
⊢ 2 ≤ 1 | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 2 ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply le_trans _ h | case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 2 ≤ 1 | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 2 ≤ index (MonoidHom.ker (MulAc... | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [← hH, ← Subgroup.normalCore_eq_ker H] | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 2 ≤ index (MonoidHom.ker (MulAc... | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ index H ≤ index (normalCore H) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | apply Nat.le_of_dvd | α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ index H ≤ index (normalCore H) | case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 0 < index (normalCore H)... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulActio... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | exact Subgroup.index_dvd_of_le (Subgroup.normalCore_le H) | case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ index H ∣ index (normalC... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (M... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [zero_lt_iff, Subgroup.normalCore_eq_ker H, Subgroup.index_ker f,
Nat.card_eq_fintype_card] | case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ 0 < index (normalCore H) | case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ Fintype.card ↑(Set.range... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (M... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | exact Fintype.card_ne_zero | case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (MulAction.toPermHom G (G ⧸ H))) ≤ 1
⊢ Fintype.card ↑(Set.range... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this✝ : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
this : index H ≠ 0
h : index (MonoidHom.ker (M... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | rw [hH] | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ index H ≠ 0 | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ 2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ index H ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Subgroup.normal_of_index_eq_two | [160, 1] | [200, 22] | norm_num | case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ 2 ≠ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type ?u.28021
inst✝² : Fintype α
inst✝¹ : DecidableEq α
G : Type u_1
inst✝ : Group G
H : Subgroup G
hH : index H = 2
this : Fintype (G ⧸ H)
f : G →* Equiv.Perm (G ⧸ H) := MulAction.toPermHom G (G ⧸ H)
⊢ 2 ≠ 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | induction' g using Equiv.Perm.swap_induction_on with f x y hxy hf | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ g = List.prod l | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ 1 = List.prod l
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
hf : ∃ l, (∀ s ∈ l, IsSwap s) ∧ f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ g = List.prod l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | use List.nil | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ 1 = List.prod l | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ (∀ s ∈ [], IsSwap s) ∧ 1 = List.prod [] | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ 1 = List.prod l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | constructor | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ (∀ s ∈ [], IsSwap s) ∧ 1 = List.prod [] | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∀ s ∈ [], IsSwap s
case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ 1 = List.prod [] | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ (∀ s ∈ [], IsSwap s) ∧ 1 = List.prod []
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | intro s hs | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∀ s ∈ [], IsSwap s | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ IsSwap s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ ∀ s ∈ [], IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | exfalso | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ IsSwap s | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | exact List.not_mem_nil s hs | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Perm α
hs : s ∈ []
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | simp only [List.prod_nil] | case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ 1 = List.prod [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
⊢ 1 = List.prod []
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | obtain ⟨l, hl, hf⟩ := hf | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
hf : ∃ l, (∀ s ∈ l, IsSwap s) ∧ f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
hf : ∃ l, (∀ s ∈ l, IsSwap s) ∧ f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | use Equiv.swap x y::l | case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ (∀ s ∈ swap x y :: l, IsSwap s) ∧ swap x y * f = List.prod (swap x y :: l) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∃ l, (∀ s ∈ l, IsSwap s) ∧ swap x y * f = List.prod l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | constructor | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ (∀ s ∈ swap x y :: l, IsSwap s) ∧ swap x y * f = List.prod (swap x y :: l) | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∀ s ∈ swap x y :: l, IsSwap s
case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ (∀ s ∈ swap x y :: l, IsSwap s) ∧ swap x y * f = List.prod (swap x y :: l)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | rw [List.prod_cons] | case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ swap x y * f = List.prod (swap x y :: l) | case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ swap x y * f = swap x y * List.prod l | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ swap x y * f = List.prod (swap x y :: l)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | rw [hf] | case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ swap x y * f = swap x y * List.prod l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ swap x y * f = swap x y * List.prod l
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | intro s hs | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∀ s ∈ swap x y :: l, IsSwap s | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ swap x y :: l
⊢ IsSwap s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
⊢ ∀ s ∈ swap x y :: l, IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | rw [List.mem_cons] at hs | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ swap x y :: l
⊢ IsSwap s | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y ∨ s ∈ l
⊢ IsSwap s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ swap x y :: l
⊢ IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | cases' hs with hs hs | case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y ∨ s ∈ l
⊢ IsSwap s | case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap s
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y ∨ s ∈ l
⊢ IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | rw [hs] | case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap s
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm ... | case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap (swap x y)
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : Li... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap s
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ :... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | exact ⟨x, y, hxy, rfl⟩ | case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap (swap x y)
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : Li... | case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ l
⊢ IsSwap s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s = swap x y
⊢ IsSwap (swap x y)
case h.left.inr
α : Type u_1
inst✝¹ : Fintype ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | Equiv.Perm.is_prod_swap_list | [206, 1] | [222, 12] | exact hl s hs | case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ l
⊢ IsSwap s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
f : Perm α
x y : α
hxy : x ≠ y
l : List (Perm α)
hl : ∀ s ∈ l, IsSwap s
hf : f = List.prod l
s : Perm α
hs : s ∈ l
⊢ IsSwap s
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have hG' := Subgroup.normal_of_index_eq_two hG | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
⊢ alternatingGroup α = G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
⊢ alternatingGroup α = G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
⊢ alternatingGroup α = G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | let s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
⊢ alternatingGroup α = G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ alternatingGroup α = G | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
⊢ alternatingGroup α = G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [alternatingGroup_eq_sign_ker, ← QuotientGroup.ker_mk' G] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ alternatingGroup α = G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ MonoidHom.ker Equiv.Perm.sign = MonoidHom.ker (QuotientGroup.mk' G) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ alternatingGroup α = G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | ext g | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ MonoidHom.ker Equiv.Perm.sign = MonoidHom.ker (QuotientGroup.mk' G) | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ g ∈ MonoidHom.ker Equiv.Perm.sign ↔ g ∈ MonoidHom.ker (QuotientGroup.mk' G) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
⊢ MonoidHom.ker Equiv.Perm.sign = MonoidHom.ker (QuotientGroup.mk' ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [Equiv.Perm.sign.mem_ker, (QuotientGroup.mk' G).mem_ker] | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ g ∈ MonoidHom.ker Equiv.Perm.sign ↔ g ∈ MonoidHom.ker (QuotientGroup.mk' G) | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ Equiv.Perm.sign g = 1 ↔ (QuotientGroup.mk' G) g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ g ∈ MonoidHom.ker Equiv.Perm.sign ↔ g ∈ M... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have h2 : Fact (Nat.Prime 2) := Fact.mk Nat.prime_two | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ Equiv.Perm.sign g = 1 ↔ (QuotientGroup.mk' G) g = 1 | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Equiv.Perm.sign g = 1 ↔ (QuotientGroup.mk' G) g = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
⊢ Equiv.Perm.sign g = 1 ↔ (QuotientGroup.mk... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have : ∃ g : Equiv.Perm α, g.IsSwap ∧ g ∉ G := by
by_contra h; push_neg at h
suffices G = ⊤ by
rw [this, Subgroup.index_top] at hG
norm_num at hG
rw [eq_top_iff, ← Equiv.Perm.closure_isSwap, Subgroup.closure_le G]
intro g hg
simp only [Set.mem_setOf_eq] at hg
simp only [SetLike.mem_coe]
exact h g ... | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
⊢ Equiv.Perm.sign g = 1 ↔ (QuotientG... | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
this : ∃ g, Equiv.Perm.IsSwap g ∧ g ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutat... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨k, hk, hk'⟩ := this | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
this : ∃ g, Equiv.Perm.IsSwap g ∧ g ... | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutat... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have this' : ∀ g : Equiv.Perm α, g.IsSwap → s g = s k := by
intro g hg
obtain ⟨a, b, hab, habg⟩ := hg
obtain ⟨x, y, hxy, hxyk⟩ := hk
obtain ⟨u, hu⟩ := Equiv.Perm.isConj_swap hab hxy
let hu' := congr_arg s (SemiconjBy.eq hu)
simp only [map_mul] at hu'
apply mul_left_cancel (a := s u)
rw [habg, hxyk, hu']... | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have hsk2 : s k ^ 2 = 1 := by
rw [pow_two]; rw [← map_mul]
obtain ⟨x, y, _, hxyk⟩ := hk
rw [hxyk]
rw [Equiv.swap_mul_self]
rw [map_one] | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨l, hl, hg⟩ := g.is_prod_swap_list | case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | let hsg := Equiv.Perm.sign_prod_list_swap hl | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prim... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [← hg] at hsg | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prim... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | have hsg' : s g = s k ^ l.length := by
rw [hg]
rw [map_list_prod]
rw [List.prod_eq_pow_card (List.map s l) (s k) _]
rw [List.length_map]
intro x hx
simp only [List.mem_map] at hx
obtain ⟨y, hyl, hxy⟩ := hx
rw [← hxy]
apply this'; exact hl y hyl | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prim... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨m, hm⟩ := Nat.even_or_odd' l.length | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Pe... | case h.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Eq... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prim... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | cases' hm with hm hm | case h.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Eq... | case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Na... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | refine' isCommutative_of_prime_order (hp := h2) _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Std.Commutative fun x x_1 => x * x_1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Fintype.card (Equiv.Perm α ⧸ G) = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Std.Commutative fun x x_... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [← Nat.card_eq_fintype_card] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Fintype.card (Equiv.Perm α ⧸ G) = 2 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Nat.card (Equiv.Perm α ⧸ G) = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Fintype.card (Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | exact hG | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Nat.card (Equiv.Perm α ⧸ G) = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
⊢ Nat.card (Equiv.Perm α ⧸... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | by_contra h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
⊢ ∃ g, Equiv.Perm.IsSwap g ∧ g ∉ G | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ¬∃ g, Equiv.Perm.IsSwap g ∧ g ∉ G
⊢ Fal... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | push_neg at h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ¬∃ g, Equiv.Perm.IsSwap g ∧ g ∉ G
⊢ Fal... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | suffices G = ⊤ by
rw [this, Subgroup.index_top] at hG
norm_num at hG | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [eq_top_iff, ← Equiv.Perm.closure_isSwap, Subgroup.closure_le G] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | intro g hg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [Set.mem_setOf_eq] at hg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [SetLike.mem_coe] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | exact h g hg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwa... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [this, Subgroup.index_top] at hG | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : 1 = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap g → g ∈ G
this... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | norm_num at hG | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : 1 = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
h : ∀ (g : Equiv.Perm α), Equiv.Perm.IsSwap g → g ∈ G
this... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : 1 = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | intro g hg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨a, b, hab, habg⟩ := hg | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
... | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨x, y, hxy, hxyk⟩ := hk | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk... | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨u, hu⟩ := Equiv.Perm.isConj_swap hab hxy | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k... | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fac... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | let hu' := congr_arg s (SemiconjBy.eq hu) | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [map_mul] at hu' | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | apply mul_left_cancel (a := s u) | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [habg, hxyk, hu'] | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | apply hG''.comm | case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x *... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g✝ : Equiv.Perm α
h2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [pow_two] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [← map_mul] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨x, y, _, hxyk⟩ := hk | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [hxyk] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [Equiv.swap_mul_self] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [map_one] | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk'... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [hg] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [map_list_prod] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [List.prod_eq_pow_card (List.map s l) (s k) _] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [List.length_map] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | intro x hx | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [List.mem_map] at hx | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | obtain ⟨y, hyl, hxy⟩ := hx | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equiv.Perm.IsSwap k
h... | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equi... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | rw [← hxy] | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equi... | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equi... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : St... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | apply this' | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Equi... | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : St... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | exact hl y hyl | case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k : Equiv.Perm α
hk : Eq... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [hm, pow_mul, hsk2, Int.units_sq, one_pow] at hsg hsg' | case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [hsg, hsg'] | case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [hm, pow_add, pow_mul, pow_one, hsk2, Int.units_sq, one_pow, one_mul] at hsg hsg' | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [hsg, hsg', neg_units_ne_self, false_iff, ne_eq] | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | simp only [QuotientGroup.mk'_apply, QuotientGroup.eq_one_iff, s] | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | is_alternating_of_index_2 | [226, 1] | [286, 14] | exact hk' | case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact (Nat.Prime 2)
hG'' : Std.Commutative fun x x_1 => x * x_1
k ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G = 2
hG' : Subgroup.Normal G
s : Equiv.Perm α →* Equiv.Perm α ⧸ G := QuotientGroup.mk' G
g : Equiv.Perm α
h2 : Fact... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | cases' Nat.eq_zero_or_pos G.index with h h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
⊢ alternatingGroup α ≤ G | case inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
h : Subgroup.index G = 0
⊢ alternatingGroup α ≤ G
case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α)... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
⊢ alternatingGroup α ≤ G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | cases' eq_or_gt_of_le (Nat.succ_le_iff.mpr h) with h h | case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
h : Subgroup.index G > 0
⊢ alternatingGroup α ≤ G | case inr.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
h✝ : Subgroup.index G > 0
h : Subgroup.index G = Nat.succ 0
⊢ alternatingGroup α ≤ G
case inr.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
h : Subgroup.index G > 0
⊢ alternatingGroup α ≤ G
TACTIC:
|
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