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/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | intro i hi | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
⊢ ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i < Fintype.card α + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
⊢ ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [Nat.lt_succ_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i < Fintype.card α + 1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ 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
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i < Fintype.card α + 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply le_trans _ (Finset.card_le_univ g.support) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ Fintype.card α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ (Equiv.Perm.support g).card | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply Equiv.Perm.le_card_support_of_mem_cycleType | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ (Equiv.Perm.support g).card | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ∈ Equiv.Perm.cycleType g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ≤ (Equiv.Perm.support g).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact hi | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ∈ Equiv.Perm.cycleType g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType g
⊢ i ∈ Equiv.Perm.cycleType g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp_rw [hlc] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | fsc a = i})) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod Finset.univ fun x => Nat.factorial (Multiset.count (↑x) (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | fsc a = i})) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [← Finset.prod_range fun i => (g.cycleType.count i).factorial] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod Finset.univ fun x => Nat.factorial (Multiset.count (↑x) (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod Finset.univ fun x => Nat.factorial (Multiset.count (↑x) (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [← Multiset.prod_toFinset] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply symm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))) =
Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))) =
Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply Finset.prod_subset_one_on_sdiff | case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))) =
Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)) ⊆ Finset.range (Fintype.card α + 1)
case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) = 1
case a.hfg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) =
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ (Finset.prod (Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))) fun n =>
Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g))) =
Finset.prod (Finset.range (Fintype.card α + 1)) fun i => Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g))
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact g.cycleType.nodup_dedup | case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hm
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.Nodup (Multiset.dedup (Equiv.Perm.cycleType g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | intro i hi | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)) ⊆ Finset.range (Fintype.card α + 1) | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Finset.range (Fintype.card α + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)) ⊆ Finset.range (Fintype.card α + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [Finset.mem_range] | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Finset.range (Fintype.card α + 1) | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i < Fintype.card α + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Finset.range (Fintype.card α + 1)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply hl_lt | case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i < Fintype.card α + 1 | case a.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Equiv.Perm.cycleType g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i < Fintype.card α + 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simpa only [Multiset.mem_toFinset, Multiset.mem_dedup] using hi | case a.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Equiv.Perm.cycleType g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ i ∈ Equiv.Perm.cycleType g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | intro i hi | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) = 1 | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Finset.mem_sdiff, Finset.mem_range, Multiset.mem_toFinset, Multiset.mem_dedup] at hi | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1 | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i ∈ Finset.range (Fintype.card α + 1) \ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g))
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [Multiset.count_eq_zero_of_not_mem hi.2] | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1 | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial 0 = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial (Multiset.count i (Equiv.Perm.cycleType g)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact Nat.factorial_zero | case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial 0 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
i : ℕ
hi : i < Fintype.card α + 1 ∧ i ∉ Equiv.Perm.cycleType g
⊢ Nat.factorial 0 = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact fun i _ ↦ rfl | case a.hfg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) =
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.hfg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
hlc : ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
hl_lt : ∀ i ∈ Equiv.Perm.cycleType g, i < Fintype.card α + 1
⊢ ∀ x ∈ Multiset.toFinset (Multiset.dedup (Equiv.Perm.cycleType g)),
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g)) =
Nat.factorial (Multiset.count x (Equiv.Perm.cycleType g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [fsc, Fin.mk.injEq, Set.mem_setOf_eq] at hx' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
⊢ x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
⊢ x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Function.comp_apply, Set.mem_setOf_eq] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact ⟨hx, hx'⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
x✝ : ↑{x | fsc x = { val := i, isLt := hi }}
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Bijective u | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Injective u
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Surjective u | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Bijective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rintro ⟨x, hx⟩ | case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Surjective u | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
⊢ ∃ a, u a = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Surjective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Function.comp_apply, Set.mem_setOf_eq] at hx | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
⊢ ∃ a, u a = { val := x, property := hx } | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ ∃ a, u a = { val := x, property := hx } | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
⊢ ∃ a, u a = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | use! x | case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ ∃ a, u a = { val := x, property := hx } | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ?property } ∈ {x | fsc x = { val := i, isLt := hi }} | Please generate a tactic in lean4 to solve the state.
STATE:
case right.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ ∃ a, u a = { val := x, property := hx }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact hx.1 | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ?property } ∈ {x | fsc x = { val := i, isLt := hi }} | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ⋯ } ∈ {x | fsc x = { val := i, isLt := hi }} | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ x ∈ Equiv.Perm.cycleFactorsFinset g
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ?property } ∈ {x | fsc x = { val := i, isLt := hi }}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [fsc, Fin.mk.injEq, Set.mem_setOf_eq] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ⋯ } ∈ {x | fsc x = { val := i, isLt := hi }} | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ (Equiv.Perm.support x).card = i | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ { val := x, property := ⋯ } ∈ {x | fsc x = { val := i, isLt := hi }}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | exact hx.2 | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ (Equiv.Perm.support x).card = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i
⊢ (Equiv.Perm.support x).card = i
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rintro ⟨⟨x, hx⟩, hx'⟩ ⟨⟨y, hy⟩, hy'⟩ h | case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Injective u | case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h :
u { val := { val := x, property := hx }, property := hx' } =
u { val := { val := y, property := hy }, property := hy' }
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' } | Please generate a tactic in lean4 to solve the state.
STATE:
case left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Function.Injective u
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Function.comp_apply, Set.coe_setOf, Set.mem_setOf_eq, Subtype.mk.injEq, u] at h | case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h :
u { val := { val := x, property := hx }, property := hx' } =
u { val := { val := y, property := hy }, property := hy' }
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' } | case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h : x = y
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' } | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h :
u { val := { val := x, property := hx }, property := hx' } =
u { val := { val := y, property := hy }, property := hy' }
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [h, Set.coe_setOf, Set.mem_setOf_eq, Subtype.mk.injEq] | case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h : x = y
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case left.mk.mk.mk.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
x : Equiv.Perm α
hx : x ∈ Equiv.Perm.cycleFactorsFinset g
hx' : { val := x, property := hx } ∈ {x | fsc x = { val := i, isLt := hi }}
y : Equiv.Perm α
hy : y ∈ Equiv.Perm.cycleFactorsFinset g
hy' : { val := y, property := hy } ∈ {x | fsc x = { val := i, isLt := hi }}
h : x = y
⊢ { val := { val := x, property := hx }, property := hx' } = { val := { val := y, property := hy }, property := hy' }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) ↔
∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) →
∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ (∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t) →
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) ↔
∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | intro hz | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) →
∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) →
∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [Subgroup.mem_map] at hz | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | obtain ⟨⟨k, hkK⟩, hk, hk'⟩ := hz | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk : { val := k, property := hkK } ∈ MonoidHom.ker (φ g)
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
hz :
∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [MonoidHom.mem_ker] at hk | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk : { val := k, property := hkK } ∈ MonoidHom.ker (φ g)
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk : { val := k, property := hkK } ∈ MonoidHom.ker (φ g)
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | intro t ht | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Commute z t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
⊢ ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | change z * t = t * z | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Commute z t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ z * t = t * z | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Commute z t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [← mul_inv_eq_iff_eq_mul, ← MulAut.conj_apply, ← ConjAct.ofConjAct_toConjAct z,
← ConjAct.smul_eq_mulAut_conj _ t, ← hk'] | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ z * t = t * z | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } • t = t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ z * t = t * z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [Subgroup.coeSubtype, Subgroup.coe_mk] | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } • t = t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ k • t = t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } • t = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [← φ_eq g k hkK t ht, hk] | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ k • t = t | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := t, property := ht }) = t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ k • t = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rfl | case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := t, property := ht }) = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
k : ConjAct (Equiv.Perm α)
hkK : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hk' :
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := k, property := hkK } =
ConjAct.toConjAct z
hk : (φ g) { val := k, property := hkK } = 1
t : Equiv.Perm α
ht : t ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := t, property := ht }) = t
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | intro H | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ (∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t) →
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
⊢ (∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t) →
ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [Subgroup.mem_map] | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g)) | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈
Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) (MonoidHom.ker (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | use! ConjAct.toConjAct z | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ { val := ConjAct.toConjAct z, property := ?property } ∈ MonoidHom.ker (φ g) ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g))
{ val := ConjAct.toConjAct z, property := ?property } =
ConjAct.toConjAct z | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ∃ x ∈ MonoidHom.ker (φ g),
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) x = ConjAct.toConjAct z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [MonoidHom.mem_ker] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ { val := ConjAct.toConjAct z, property := ⋯ } ∈ MonoidHom.ker (φ g) ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1 ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ { val := ConjAct.toConjAct z, property := ⋯ } ∈ MonoidHom.ker (φ g) ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | constructor | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1 ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z | case h.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1
case h.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1 ∧
(Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [MulAction.mem_stabilizer_iff] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z • g = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [ConjAct.smul_eq_mulAut_conj] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z • g = g | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (MulAut.conj (ConjAct.ofConjAct (ConjAct.toConjAct z))) g = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.toConjAct z • g = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [MulAut.conj_apply] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (MulAut.conj (ConjAct.ofConjAct (ConjAct.toConjAct z))) g = g | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g * (ConjAct.ofConjAct (ConjAct.toConjAct z))⁻¹ = g | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (MulAut.conj (ConjAct.ofConjAct (ConjAct.toConjAct z))) g = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [mul_inv_eq_iff_eq_mul] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g * (ConjAct.ofConjAct (ConjAct.toConjAct z))⁻¹ = g | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g = g * ConjAct.ofConjAct (ConjAct.toConjAct z) | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g * (ConjAct.ofConjAct (ConjAct.toConjAct z))⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [ConjAct.ofConjAct_toConjAct] | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g = g * ConjAct.ofConjAct (ConjAct.toConjAct z) | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ z * g = g * z | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ ConjAct.ofConjAct (ConjAct.toConjAct z) * g = g * ConjAct.ofConjAct (ConjAct.toConjAct z)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | exact Equiv.Perm.commute_of_mem_cycleFactorsFinset_commute z g H | case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ z * g = g * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case property
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ z * g = g * z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | ext ⟨c, hc⟩ | case h.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1 | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ ↑(((φ g) { val := ConjAct.toConjAct z, property := ⋯ }) { val := c, property := hc }) x✝ =
↑(1 { val := c, property := hc }) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (φ g) { val := ConjAct.toConjAct z, property := ⋯ } = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [φ_eq'] | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ ↑(((φ g) { val := ConjAct.toConjAct z, property := ⋯ }) { val := c, property := hc }) x✝ =
↑(1 { val := c, property := hc }) x✝ | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (z * c * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ ↑(((φ g) { val := ConjAct.toConjAct z, property := ⋯ }) { val := c, property := hc }) x✝ =
↑(1 { val := c, property := hc }) x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | rw [H c hc] | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (z * c * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝ | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (c * z * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (z * c * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [mul_inv_cancel_right, Equiv.Perm.coe_one, id.def, Subtype.coe_mk] | case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (c * z * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.left.H.mk.a.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x✝ : α
⊢ (c * z * z⁻¹) x✝ = ↑(1 { val := c, property := hc }) x✝
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_ker_iff | [2093, 1] | [2124, 55] | simp only [Subgroup.coeSubtype, Subgroup.coe_mk] | case h.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g z : Equiv.Perm α
H : ∀ t ∈ Equiv.Perm.cycleFactorsFinset g, Commute z t
⊢ (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) { val := ConjAct.toConjAct z, property := ⋯ } =
ConjAct.toConjAct z
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | induction s using Finset.induction_on with
| empty => simp only [Finset.noncommProd_empty]
| insert ha hs =>
rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ ha]
rw [hf _ (Finset.mem_insert_self _ _), one_mul]
apply hs
intro a ha
exact hf _ (Finset.mem_insert_of_mem ha) | α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
s : Finset α
f : α → β
comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)
hf : ∀ a ∈ s, f a = 1
⊢ Finset.noncommProd s f comm = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
s : Finset α
f : α → β
comm : Set.Pairwise ↑s fun a b => Commute (f a) (f b)
hf : ∀ a ∈ s, f a = 1
⊢ Finset.noncommProd s f comm = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | simp only [Finset.noncommProd_empty] | case empty
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
comm : Set.Pairwise ↑∅ fun a b => Commute (f a) (f b)
hf : ∀ a ∈ ∅, f a = 1
⊢ Finset.noncommProd ∅ f comm = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
comm : Set.Pairwise ↑∅ fun a b => Commute (f a) (f b)
hf : ∀ a ∈ ∅, f a = 1
⊢ Finset.noncommProd ∅ f comm = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ ha] | case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ Finset.noncommProd (insert a✝ s✝) f comm = 1 | case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ f a✝ * Finset.noncommProd s✝ f ⋯ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ Finset.noncommProd (insert a✝ s✝) f comm = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | rw [hf _ (Finset.mem_insert_self _ _), one_mul] | case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ f a✝ * Finset.noncommProd s✝ f ⋯ = 1 | case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ Finset.noncommProd s✝ f ⋯ = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ f a✝ * Finset.noncommProd s✝ f ⋯ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | apply hs | case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ Finset.noncommProd s✝ f ⋯ = 1 | case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ ∀ a ∈ s✝, f a = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ Finset.noncommProd s✝ f ⋯ = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | intro a ha | case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ ∀ a ∈ s✝, f a = 1 | case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha✝ : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
a : α
ha : a ∈ s✝
⊢ f a = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
⊢ ∀ a ∈ s✝, f a = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Finset.noncommProd_eq_one | [2146, 1] | [2159, 47] | exact hf _ (Finset.mem_insert_of_mem ha) | case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha✝ : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
a : α
ha : a ∈ s✝
⊢ f a = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.hf
α✝ : Type ?u.304426
inst✝³ : DecidableEq α✝
inst✝² : Fintype α✝
g : Equiv.Perm α✝
α : Type u_3
inst✝¹ : DecidableEq α
β : Type u_4
inst✝ : Monoid β
f : α → β
a✝ : α
s✝ : Finset α
ha✝ : a✝ ∉ s✝
hs : ∀ (comm : Set.Pairwise ↑s✝ fun a b => Commute (f a) (f b)), (∀ a ∈ s✝, f a = 1) → Finset.noncommProd s✝ f comm = 1
comm : Set.Pairwise ↑(insert a✝ s✝) fun a b => Commute (f a) (f b)
hf : ∀ a ∈ insert a✝ s✝, f a = 1
a : α
ha : a ∈ s✝
⊢ f a = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | Equiv.Perm.cycleOf_ne_one_iff_mem | [2161, 1] | [2164, 46] | rw [Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff, Equiv.Perm.mem_support,
ne_eq, Equiv.Perm.cycleOf_eq_one_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x : α
⊢ Equiv.Perm.cycleOf g x ≠ 1 ↔ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
x : α
⊢ Equiv.Perm.cycleOf g x ≠ 1 ↔ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_not_mem_cycleFactorsFinset | [2173, 1] | [2176, 24] | rw [θAux, dif_neg hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k v x = (Equiv.Perm.ofSubtype k) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k v x = (Equiv.Perm.ofSubtype k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_mem_fixedPoints | [2178, 1] | [2183, 11] | rw [θAux, dif_neg] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ θAux g k v x = (Equiv.Perm.ofSubtype k) x | case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ θAux g k v x = (Equiv.Perm.ofSubtype k) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_mem_fixedPoints | [2178, 1] | [2183, 11] | rw [Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff, Equiv.Perm.not_mem_support] | case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g | case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ g x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_mem_fixedPoints | [2178, 1] | [2183, 11] | exact hx | case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ g x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hnc
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ g x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_mem_fixedPoints_mem | [2185, 1] | [2189, 19] | rw [θAux_apply_of_mem_fixedPoints hx, Equiv.Perm.ofSubtype_apply_of_mem k hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ θAux g k v x ∈ Function.fixedPoints ⇑g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ ↑(k { val := x, property := hx }) ∈ Function.fixedPoints ⇑g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ θAux g k v x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_mem_fixedPoints_mem | [2185, 1] | [2189, 19] | exact (k _).prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ ↑(k { val := x, property := hx }) ∈ Function.fixedPoints ⇑g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : x ∈ Function.fixedPoints ⇑g
⊢ ↑(k { val := x, property := hx }) ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq | [2191, 1] | [2196, 36] | suffices c = ⟨g.cycleOf x, ?_⟩ by
rw [this, θAux, dif_pos]
rw [hx]; exact c.prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ θAux g k v x = ↑(v c) x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ c = { val := Equiv.Perm.cycleOf g x, property := ⋯ } | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ θAux g k v x = ↑(v c) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq | [2191, 1] | [2196, 36] | simp only [← Subtype.coe_inj, hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ c = { val := Equiv.Perm.cycleOf g x, property := ⋯ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ c = { val := Equiv.Perm.cycleOf g x, property := ⋯ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq | [2191, 1] | [2196, 36] | rw [this, θAux, dif_pos] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
this : c = { val := Equiv.Perm.cycleOf g x, property := ?m.315200 }
⊢ θAux g k v x = ↑(v c) x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
this : c = { val := Equiv.Perm.cycleOf g x, property := ?m.315200 }
⊢ θAux g k v x = ↑(v c) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq | [2191, 1] | [2196, 36] | rw [hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ ↑c ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq | [2191, 1] | [2196, 36] | exact c.prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ ↑c ∈ Equiv.Perm.cycleFactorsFinset g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ ↑c ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq_mem | [2198, 1] | [2205, 85] | rw [θAux_apply_of_cycleOf_eq c hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = ↑c | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq_mem | [2198, 1] | [2205, 85] | obtain ⟨m, hm⟩ := (v c).prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : (fun x => ↑c ^ x) m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq_mem | [2198, 1] | [2205, 85] | dsimp only at hm | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : (fun x => ↑c ^ x) m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : (fun x => ↑c ^ x) m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq_mem | [2198, 1] | [2205, 85] | rw [← hm, ← hx] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g ((Equiv.Perm.cycleOf g x ^ m) x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g (↑(v c) x) = ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_apply_of_cycleOf_eq_mem | [2198, 1] | [2205, 85] | simp only [Equiv.Perm.cycleOf_zpow_apply_self, Equiv.Perm.cycleOf_self_apply_zpow] | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g ((Equiv.Perm.cycleOf g x ^ m) x) = Equiv.Perm.cycleOf g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : Equiv.Perm.cycleOf g x = ↑c
m : ℤ
hm : ↑c ^ m = ↑(v c)
⊢ Equiv.Perm.cycleOf g ((Equiv.Perm.cycleOf g x ^ m) x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | by_cases hx : g.cycleOf x ∈ g.cycleFactorsFinset | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | rw [θAux_apply_of_cycleOf_eq ⟨g.cycleOf x, hx⟩ rfl] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | obtain ⟨m, hm⟩ := (v _).prop | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : (fun x => ↑?m.317682 ^ x) m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | dsimp only at hm | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : (fun x => ↑?m.317682 ^ x) m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑?m.317682 ^ m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : (fun x => ↑?m.317682 ^ x) m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | rw [← hm] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑?m.317682 ^ m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m = ↑(v { val := Equiv.Perm.cycleOf g x, property := hx })
⊢ Equiv.Perm.cycleOf g ((↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m) x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑?m.317682 ^ m = ↑(v ?m.317682)
⊢ Equiv.Perm.cycleOf g (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | simp only [Equiv.Perm.cycleOf_zpow_apply_self, Equiv.Perm.cycleOf_self_apply_zpow] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m = ↑(v { val := Equiv.Perm.cycleOf g x, property := hx })
⊢ Equiv.Perm.cycleOf g ((↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m) x) = Equiv.Perm.cycleOf g x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
m : ℤ
hm : ↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m = ↑(v { val := Equiv.Perm.cycleOf g x, property := hx })
⊢ Equiv.Perm.cycleOf g ((↑{ val := Equiv.Perm.cycleOf g x, property := hx } ^ m) x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | rw [g.cycleOf_mem_cycleFactorsFinset_iff, Equiv.Perm.not_mem_support] at hx | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | rw [g.cycleOf_eq_one_iff.mpr hx, g.cycleOf_eq_one_iff,
← Function.mem_fixedPoints_iff] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ θAux g k v x ∈ Function.fixedPoints ⇑g | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ Equiv.Perm.cycleOf g (θAux g k v x) = Equiv.Perm.cycleOf g x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | apply θAux_apply_of_mem_fixedPoints_mem | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ θAux g k v x ∈ Function.fixedPoints ⇑g | case neg.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ θAux g k v x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_cycleOf_apply_eq | [2207, 1] | [2219, 13] | exact hx | case neg.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_one | [2221, 1] | [2226, 51] | unfold θAux | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
⊢ θAux g 1 1 x = x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
⊢ (if hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g then
↑(1 { val := Equiv.Perm.cycleOf g x, property := hx }) x
else (Equiv.Perm.ofSubtype 1) x) =
x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
⊢ θAux g 1 1 x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_one | [2221, 1] | [2226, 51] | split_ifs | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
⊢ (if hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g then
↑(1 { val := Equiv.Perm.cycleOf g x, property := hx }) x
else (Equiv.Perm.ofSubtype 1) x) =
x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := Equiv.Perm.cycleOf g x, property := h✝ }) x = x
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ (Equiv.Perm.ofSubtype 1) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
⊢ (if hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g then
↑(1 { val := Equiv.Perm.cycleOf g x, property := hx }) x
else (Equiv.Perm.ofSubtype 1) x) =
x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_one | [2221, 1] | [2226, 51] | simp only [Pi.one_apply, OneMemClass.coe_one, Equiv.Perm.coe_one, id_eq] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := Equiv.Perm.cycleOf g x, property := h✝ }) x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(1 { val := Equiv.Perm.cycleOf g x, property := h✝ }) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_one | [2221, 1] | [2226, 51] | simp only [map_one, Equiv.Perm.coe_one, id_eq] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ (Equiv.Perm.ofSubtype 1) x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
x : α
h✝ : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ (Equiv.Perm.ofSubtype 1) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_mul | [2228, 1] | [2248, 65] | by_cases hx : g.cycleOf x ∈ g.cycleFactorsFinset | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∉ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.θAux_mul | [2228, 1] | [2248, 65] | rw [θAux_apply_of_cycleOf_eq ⟨g.cycleOf x, hx⟩ (θAux_apply_of_cycleOf_eq_mem ⟨_, hx⟩ rfl),
θAux_apply_of_cycleOf_eq ⟨g.cycleOf x, hx⟩ rfl,
θAux_apply_of_cycleOf_eq ⟨g.cycleOf x, hx⟩ rfl] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ ↑(v' { val := Equiv.Perm.cycleOf g x, property := hx }) (↑(v { val := Equiv.Perm.cycleOf g x, property := hx }) x) =
↑((v' * v) { val := Equiv.Perm.cycleOf g x, property := hx }) x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
k' : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v' : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
k : Equiv.Perm ↑(Function.fixedPoints ⇑g)
v : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
x : α
hx : Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
⊢ θAux g k' v' (θAux g k v x) = θAux g (k' * k) (v' * v) x
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.