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 | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | rw [← Nat.succ_le_iff] at h | case inr.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
h : Nat.succ 0 < Subgroup.index G
⊢ alternatingGroup α ≤ G | case inr.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
h : Nat.succ (Nat.succ 0) ≤ Subgroup.index G
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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
h : Nat.succ 0 < Subgroup.index 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] | norm_num at h | case inr.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
h : Nat.succ (Nat.succ 0) ≤ Subgroup.index G
⊢ alternatingGroup α ≤ G | case inr.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
h : 2 ≤ Subgroup.index G
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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
h : Nat.succ (Nat.succ 0) ≤ Subgroup.index 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] | apply le_of_eq | case inr.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
h : 2 ≤ Subgroup.index G
⊢ alternatingGroup α ≤ G | case inr.inr.a
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 : 2 ≤ Subgroup.index G
⊢ alternatingGroup α = G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.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
h : 2 ≤ Subgroup.index 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] | apply is_alternating_of_index_2 | case inr.inr.a
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 : 2 ≤ Subgroup.index G
⊢ alternatingGroup α = G | case inr.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G = 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a
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 : 2 ≤ Subgroup.index 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] | refine' le_antisymm _ h | case inr.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G = 2 | case inr.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G ≤ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | refine' Nat.le_of_mul_le_mul_left _ _ | case inr.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G ≤ 2 | case inr.inr.a.hG.refine'_1
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 : 2 ≤ Subgroup.index G
⊢ ℕ
case inr.inr.a.hG.refine'_2
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.inr.a.hG
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 : 2 ≤ Subgroup.index G
⊢ Subgroup.index G ≤ 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | swap | case inr.inr.a.hG.refine'_1
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 : 2 ≤ Subgroup.index G
⊢ ℕ
case inr.inr.a.hG.refine'_2
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equ... | case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ ?inr.inr.a.hG.refine'_1 * Subgroup.index G ≤ ?inr.inr.a.hG.refine'_1 * 2
case inr.inr.a.hG.refine'_1
... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a.hG.refine'_1
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 : 2 ≤ Subgroup.index G
⊢ ℕ
case inr.inr.a.hG.refine'_2
α : Type u... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | exact Fintype.card_pos | case inr.inr.a.hG.refine'_3
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 : 2 ≤ Subgroup.index G
⊢ 0 < Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a.hG.refine'_3
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 : 2 ≤ Subgroup.index G
⊢ 0 < Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | exfalso | 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 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
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | exact Subgroup.index_ne_zero_of_finite h | 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
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | rw [Subgroup.index_eq_one] at h | 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.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 : G = ⊤
⊢ alternatingGroup α ≤ G | Please generate a tactic in lean4 to solve the state.
STATE:
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
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | rw [h] | 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 : G = ⊤
⊢ 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 : G = ⊤
⊢ alternatingGroup α ≤ ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
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 : 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] | exact le_top | 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 : G = ⊤
⊢ alternatingGroup α ≤ ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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 : G = ⊤
⊢ alternatingGroup α ≤ ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | rw [mul_comm, Subgroup.index_mul_card, mul_comm] | case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ ?inr.inr.a.hG.refine'_1 * Subgroup.index G ≤ ?inr.inr.a.hG.refine'_1 * 2 | case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ ?inr.inr.a.hG.refine'_1 * Subgroup.index ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | large_subgroup_of_perm_contains_alternating | [288, 1] | [305, 25] | exact hG | case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.a.hG.refine'_2
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 : 2 ≤ Subgroup.index G
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | contains_alternating_of_index_le_2' | [308, 1] | [313, 34] | apply large_subgroup_of_perm_contains_alternating | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ alternatingGroup α ≤ G | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥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 | contains_alternating_of_index_le_2' | [308, 1] | [313, 34] | rw [← Subgroup.index_mul_card G] | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Subgroup.index G * Fintype.card ↥G ≤ 2 * Fintype.card ↥G | Please generate a tactic in lean4 to solve the state.
STATE:
case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Fintype.card (Equiv.Perm α) ≤ 2 * Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/IndexNormal.lean | contains_alternating_of_index_le_2' | [308, 1] | [313, 34] | apply Nat.mul_le_mul_right _ hG | case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Subgroup.index G * Fintype.card ↥G ≤ 2 * Fintype.card ↥G | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hG
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
G : Subgroup (Equiv.Perm α)
hG : Subgroup.index G ≤ 2
⊢ Subgroup.index G * Fintype.card ↥G ≤ 2 * Fintype.card ↥G
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← Equiv.Perm.lcm_cycleType] at hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : g ∈ alternatingGroup α
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : g ∈ alternatingGroup α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : g ∈ alternatingGroup α
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Equiv.Perm.mem_alternatingGroup, Equiv.Perm.sign_of_cycleType] at hg0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : g ∈ alternatingGroup α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : g ∈ alternatingGroup α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | have hg4 : g.cycleType.sum ≤ 4 := by
rw [← hα4, Equiv.Perm.sum_cycleType]
apply Finset.card_le_univ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.Perm.cycleType g = ∅ ∨ Equiv.Perm.cycleType g = {2, 2} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
⊢ Equiv.Perm.cycleTyp... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | by_cases h4 : 4 ∈ g.cycleType | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
⊢ Equiv.Perm.cycleTyp... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← hα4, Equiv.Perm.sum_cycleType] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Multiset.sum (Equiv.Perm.cycleType g) ≤ 4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ (Equiv.Perm.support g).card ≤ Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ Multis... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply Finset.card_le_univ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ (Equiv.Perm.support g).card ≤ Fintype.card α | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
⊢ (Equiv... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exfalso | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | suffices g.cycleType = {4} by
rw [this, ← Units.eq_iff] at hg0 ; norm_num at hg0 | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← Multiset.cons_erase h4] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply symm | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equ... | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Multiset.singleton_eq_cons_iff] | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply And.intro rfl | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← Multiset.cons_erase h4] at hg4 | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ E... | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (4 ::ₘ Multiset.erase (Equiv.Perm.cycle... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Multiset.sum_cons, add_le_iff_nonpos_right, le_zero_iff, Multiset.sum_eq_zero_iff] at hg4 | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (4 ::ₘ Multiset.erase (Equiv.Perm.cycle... | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.erase... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | ext x | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.erase... | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Multiset.count_zero, Multiset.count_eq_zero] | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | intro hx | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply not_le.mpr (Equiv.Perm.one_lt_of_mem_cycleType (Multiset.mem_of_mem_erase hx)) | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hg4 x hx] | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num | case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
h4 : 4 ∈ Equiv.Perm.cycleType g
hg4 : ∀ x ∈ Multiset.era... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [this, ← Units.eq_iff] at hg0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : ↑((-1) ^ (Multiset.sum {4} + Multiset.card {4})) = ↑1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num at hg0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : ↑((-1) ^ (Multiset.sum {4} + Multiset.card {4})) = ↑1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∈ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : ↑((-1) ^ (Multiset.sum {4} + Multiset.card {4})) = ↑1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Multiset.eq_replicate_card] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | intro i hi | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | have : i ∣ 2 ^ n := by
apply Nat.dvd_trans _ hg
exact Multiset.dvd_lcm hi | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Nat.dvd_prime_pow] at this | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | obtain ⟨k, hk, hlcm⟩ := this | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equ... | case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | suffices k = 1 by
rw [hlcm, this]; norm_num | case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
... | case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleTyp... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply le_antisymm | case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleTyp... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← not_lt] | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | intro hk1 | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | suffices k = 2 by
apply h4
rw [this] at hlcm ; norm_num at hlcm
rw [← hlcm]
exact hi | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | refine' le_antisymm _ hk1 | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Nat.one_le_iff_ne_zero] | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | intro hk0 | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hk0] at hlcm | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num at hlcm | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hlcm] at hi | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply Nat.lt_irrefl 1 | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exact Equiv.Perm.one_lt_of_mem_cycleType hi | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.pp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exact Nat.prime_two | case neg.pp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.pp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [this] at hg0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 :
(-1) ^
(Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) +
Multiset.card (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2)) =
1
n : ℕ
hg : Multiset.lcm (Equ... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [pow_add, pow_mul, Multiset.sum_replicate, Algebra.id.smul_eq_mul,
Multiset.card_replicate, Int.units_sq, one_mul] at hg0 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 :
(-1) ^
(Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) +
Multiset.card (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2)) =
1
n : ℕ
hg : Multiset.lcm (Equ... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 :
(-1) ^
(Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) +
Multiset.card (Multiset.replicate (Multiset.card (Equi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | have hk2 : Multiset.card g.cycleType ≤ 2 := by
rw [this] at hg4 ; rw [Multiset.sum_replicate] at hg4
apply Nat.le_of_mul_le_mul_left; rw [Nat.mul_comm 2]
exact hg4
norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleTyp... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | cases' Nat.eq_or_lt_of_le hk2 with hk2 hk1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType... | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleTyp... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [this] at hg4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replica... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleTyp... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Multiset.sum_replicate] at hg4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replica... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycl... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Multiset.replicate (Multiset.card (Equiv.Perm.cycleType g)) 2) ≤ 4
h4 : 4 ∉ Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply Nat.le_of_mul_le_mul_left | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.cycl... | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cyc... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Nat.mul_comm 2] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Pe... | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Pe... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.P... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exact hg4 | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Pe... | case hc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.P... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num | case hc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.card (Equiv.Perm.cycleType g) • 2 ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hk2] at this | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.... | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [this] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Finset.mem_univ, Multiset.replicate_succ, Multiset.replicate_one,
Multiset.empty_eq_zero, Multiset.cons_ne_zero, Multiset.insert_eq_cons, eq_self_iff_true,
false_or_iff, and_self_iff] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Multiset.replicate_zero, Multiset.cons_zero] | case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 2 2
hg0 : (-1) ^ Multiset.c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [Nat.lt_succ_iff] at hk1 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.... | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | cases' Nat.eq_or_lt_of_le hk1 with hk1 hk0 | case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.Perm.... | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exfalso | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hk1, ← Units.eq_iff] at hg0 | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num at hg0 | case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Nat.lt_one_iff] at hk0 | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hk0] at this | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate (Multiset.card (Equiv.P... | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Multis... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [this] | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Multis... | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Multis... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | left | case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Multis... | case inr.inr.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Mult... | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | simp only [Multiset.replicate_zero, Multiset.empty_eq_zero] | case inr.inr.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equiv.Perm.cycleType g = Multiset.replicate 0 2
hg0 : (-1) ^ Mult... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.cycleType g
this : Equi... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply Nat.dvd_trans _ hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exact Multiset.dvd_lcm hi | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [hlcm, this] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply h4 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [this] at hlcm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num at hlcm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← hlcm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | exact hi | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ 4
h4 : 4 ∉ Equiv.Perm.c... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Mu... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← Nat.pow_le_pow_iff_right (Nat.le_refl 2)] | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | norm_num | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | rw [← hα4, ← hlcm] | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply le_trans (Equiv.Perm.le_card_support_of_mem_cycleType hi) | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.mem_V4_of_order_two_pow | [42, 1] | [122, 24] | apply Finset.card_le_univ | case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleType g) ∣ 2 ^ n
hg4 : Multiset.sum (Equiv.Perm.cycleType g) ≤ ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
g : Equiv.Perm α
hg0 : (-1) ^ (Multiset.sum (Equiv.Perm.cycleType g) + Multiset.card (Equiv.Perm.cycleType g)) = 1
n : ℕ
hg : Multiset.lcm (Equiv.Perm.cycleT... |
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | have : Nontrivial α := by
rw [← Fintype.one_lt_card_iff_nontrivial, hα4]
norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card ↥(alternatingGroup α) = 12 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 12 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
⊢ Fintype.card ↥(alternatingGroup α) = 12
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | apply mul_right_injective₀ (_: 2 ≠ 0) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 12 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ Fintype.card ↥(alternatingGroup α) = 12
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/V4.lean | alternatingGroup.A4_card | [128, 1] | [138, 6] | dsimp | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 * Fintype.card ↥(alternatingGroup α) = 2 * 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ 2 ≠ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α = 4
this : Nontrivial α
⊢ (fun x => 2 * x) (Fintype.card ↥(alternatingGroup α)) = (fun x => 2 * x) 12
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Fintype.card α... |
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