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.hψ_range_card | [2950, 1] | [2970, 18] | rw [Finset.prod] | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Equiv.Perm.cycleFactorsFinset g) fun x => Fintype.card ↥(Subgroup.zpowers x)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.prod (Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ (Finset.prod (Equiv.Perm.cycleFactorsFinset g) fun x => Fintype.card ↥(Subgroup.zpowers x)) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | apply congr_arg | case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.prod (Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val) | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val =
Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val | Please generate a tactic in lean4 to solve the state.
STATE:
case hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.prod (Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val) =
Multiset.prod (Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | apply Multiset.map_congr rfl | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val =
Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ ∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val, Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card | Please generate a tactic in lean4 to solve the state.
STATE:
case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.map (fun x => Fintype.card ↥(Subgroup.zpowers x)) (Equiv.Perm.cycleFactorsFinset g).val =
Multiset.map (fun x => (Equiv.Perm.support x).card) (Equiv.Perm.cycleFactorsFinset g).val
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | intro x hx | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ ∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val, Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card | Please generate a tactic in lean4 to solve the state.
STATE:
case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ ∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val, Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | rw [Fintype.card_zpowers, Equiv.Perm.IsCycle.orderOf] | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Equiv.Perm.IsCycle x | Please generate a tactic in lean4 to solve the state.
STATE:
case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Fintype.card ↥(Subgroup.zpowers x) = (Equiv.Perm.support x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | simp only [Finset.mem_val, Equiv.Perm.mem_cycleFactorsFinset_iff] at hx | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Equiv.Perm.IsCycle x | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : Equiv.Perm.IsCycle x ∧ ∀ a ∈ Equiv.Perm.support x, x a = g a
⊢ Equiv.Perm.IsCycle x | Please generate a tactic in lean4 to solve the state.
STATE:
case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : x ∈ (Equiv.Perm.cycleFactorsFinset g).val
⊢ Equiv.Perm.IsCycle x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hψ_range_card | [2950, 1] | [2970, 18] | exact hx.left | case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : Equiv.Perm.IsCycle x ∧ ∀ a ∈ Equiv.Perm.support x, x a = g a
⊢ Equiv.Perm.IsCycle x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hy.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g x : Equiv.Perm α
hx : Equiv.Perm.IsCycle x ∧ ∀ a ∈ Equiv.Perm.support x, x a = g a
⊢ Equiv.Perm.IsCycle x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [Subgroup.card_eq_card_quotient_mul_card_subgroup (φ g).ker] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card (↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) ⧸ MonoidHom.ker (φ g)) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [Fintype.card_congr (QuotientGroup.quotientKerEquivRange (φ g)).toEquiv] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) ⧸ MonoidHom.ker (φ g)) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) * Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card (↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) ⧸ MonoidHom.ker (φ g)) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [← Nat.card_eq_fintype_card, hφ_range_card] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) * Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) * Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [mul_comm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) *
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) *
Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [← hψ_range_card] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) = Fintype.card ↑(Set.range ⇑(θ g)) | 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
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) =
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_stabilizer_card | [2975, 1] | [2986, 22] | rw [hψ_range_card'] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) = Fintype.card ↑(Set.range ⇑(θ 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
⊢ Fintype.card ↥(MonoidHom.ker (φ g)) = Fintype.card ↑(Set.range ⇑(θ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | ext k | α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g : G
⊢ {k | IsConj g k} = MulAction.orbit (ConjAct G) g | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ k ∈ {k | IsConj g k} ↔ k ∈ MulAction.orbit (ConjAct G) g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g : G
⊢ {k | IsConj g k} = MulAction.orbit (ConjAct G) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | simp only [Set.mem_setOf_eq, isConj_iff, MulAction.mem_orbit_iff, ConjAct.smul_def] | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ k ∈ {k | IsConj g k} ↔ k ∈ MulAction.orbit (ConjAct G) g | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) ↔ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ k ∈ {k | IsConj g k} ↔ k ∈ MulAction.orbit (ConjAct G) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | constructor | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) ↔ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k | case h.mp
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) → ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) ↔ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | rintro ⟨c, hc⟩ | case h.mp
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) → ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | case h.mp.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ c, c * g * c⁻¹ = k) → ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | use ConjAct.toConjAct c | case h.mp.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ConjAct.ofConjAct (ConjAct.toConjAct c) * g * (ConjAct.ofConjAct (ConjAct.toConjAct c))⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mp.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | simp only [hc, ConjAct.ofConjAct_toConjAct] | case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ConjAct.ofConjAct (ConjAct.toConjAct c) * g * (ConjAct.ofConjAct (ConjAct.toConjAct c))⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k c : G
hc : c * g * c⁻¹ = k
⊢ ConjAct.ofConjAct (ConjAct.toConjAct c) * g * (ConjAct.ofConjAct (ConjAct.toConjAct c))⁻¹ = k
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | rintro ⟨x, hx⟩ | case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k | case h.mpr.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
x : ConjAct G
hx : ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
⊢ ∃ c, c * g * c⁻¹ = k | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
⊢ (∃ x, ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k) → ∃ c, c * g * c⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Group.conj_class_eq_conj_orbit | [2989, 1] | [2997, 26] | use ConjAct.ofConjAct x | case h.mpr.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
x : ConjAct G
hx : ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
⊢ ∃ c, c * g * c⁻¹ = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.mpr.intro
α : Type ?u.424342
inst✝² : DecidableEq α
inst✝¹ : Fintype α
g✝ : Equiv.Perm α
G : Type u_1
inst✝ : Group G
g k : G
x : ConjAct G
hx : ConjAct.ofConjAct x * g * (ConjAct.ofConjAct x)⁻¹ = k
⊢ ∃ c, c * g * c⁻¹ = k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | classical
simp_rw [Group.conj_class_eq_conj_orbit g]
simp only [mul_assoc]
rw [mul_comm]
simp only [← mul_assoc]
rw [← Equiv.Perm.conj_stabilizer_card g]
rw [mul_comm]
rw [MulAction.card_orbit_mul_card_stabilizer_eq_card_group (ConjAct (Equiv.Perm α)) g]
rw [ConjAct.card, Fintype.card_perm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | simp_rw [Group.conj_class_eq_conj_orbit g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | simp only [mul_assoc] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | rw [mul_comm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | simp only [← mul_assoc] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | rw [← Equiv.Perm.conj_stabilizer_card g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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))) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | rw [mul_comm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (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✝ g : Equiv.Perm α
⊢ Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | rw [MulAction.card_orbit_mul_card_stabilizer_eq_card_group (ConjAct (Equiv.Perm α)) g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (ConjAct (Equiv.Perm α)) = Nat.factorial (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑(MulAction.orbit (ConjAct (Equiv.Perm α)) g) *
Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card_mul_eq | [3000, 1] | [3015, 39] | rw [ConjAct.card, Fintype.card_perm] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (ConjAct (Equiv.Perm α)) = Nat.factorial (Fintype.card α) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card (ConjAct (Equiv.Perm α)) = Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Multiset.prod_pos | [3018, 1] | [3026, 60] | induction' m using Multiset.induction with a m ih | α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
m : Multiset R
h : ∀ a ∈ m, 0 < a
⊢ 0 < Multiset.prod m | case empty
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
h : ∀ a ∈ 0, 0 < a
⊢ 0 < Multiset.prod 0
case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < Multiset.prod (a ::ₘ m) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
m : Multiset R
h : ∀ a ∈ m, 0 < a
⊢ 0 < Multiset.prod m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Multiset.prod_pos | [3018, 1] | [3026, 60] | simp | case empty
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
h : ∀ a ∈ 0, 0 < a
⊢ 0 < Multiset.prod 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
h : ∀ a ∈ 0, 0 < a
⊢ 0 < Multiset.prod 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Multiset.prod_pos | [3018, 1] | [3026, 60] | rw [Multiset.prod_cons] | case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < Multiset.prod (a ::ₘ m) | case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < a * Multiset.prod m | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < Multiset.prod (a ::ₘ m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Multiset.prod_pos | [3018, 1] | [3026, 60] | exact
mul_pos (h _ <| Multiset.mem_cons_self _ _)
(ih fun a ha => h a <| Multiset.mem_cons_of_mem ha) | case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < a * Multiset.prod m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α : Type ?u.434129
inst✝³ : DecidableEq α
inst✝² : Fintype α
g : Equiv.Perm α
R : Type u_1
inst✝¹ : StrictOrderedCommSemiring R
inst✝ : Nontrivial R
a : R
m : Multiset R
ih : (∀ a ∈ m, 0 < a) → 0 < Multiset.prod m
h : ∀ a_1 ∈ a ::ₘ m, 0 < a_1
⊢ 0 < a * Multiset.prod m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.conj_class_card | [3030, 1] | [3042, 27] | rw [← Equiv.Perm.conj_class_card_mul_eq g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_class_card | [3030, 1] | [3042, 27] | rw [Nat.div_eq_of_eq_mul_left _] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_class_card | [3030, 1] | [3042, 27] | simp only [← mul_assoc] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_class_card | [3030, 1] | [3042, 27] | rw [← Equiv.Perm.conj_stabilizer_card g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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✝ g : Equiv.Perm α
⊢ 0 < Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g) | 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
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) * Multiset.prod (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.Equiv.Perm.conj_class_card | [3030, 1] | [3042, 27] | refine' Fintype.card_pos | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ g : Equiv.Perm α
⊢ 0 < Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) 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
⊢ 0 < Fintype.card ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | split_ifs with hm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
if Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a then Nat.factorial (Fintype.card α) else 0 | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α)
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
0 | 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
m : Multiset ℕ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
if Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a then Nat.factorial (Fintype.card α) else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | obtain ⟨g, hg⟩ := Equiv.permWithCycleType_nonempty_iff.mpr hm | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | ext h | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ h ∈ Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ ↔
h ∈ Finset.filter (fun h => IsConj g h) Finset.univ | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [Finset.mem_filter, Finset.mem_univ, true_and_iff, Set.mem_toFinset, Set.mem_setOf_eq] | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ h ∈ Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ ↔
h ∈ Finset.filter (fun h => IsConj g h) Finset.univ | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj g h | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ h ∈ Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ ↔
h ∈ Finset.filter (fun h => IsConj g h) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [isConj_comm] | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj g h | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj h g | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj g h
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [Equiv.Perm.isConj_iff_cycleType_eq] | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj h g | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ IsConj h g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [Equiv.permWithCycleType, Finset.mem_filter] at hg | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g h : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
h : Equiv.Perm α
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [hg.2] | case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g h : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g h : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Equiv.Perm.cycleType h = m ↔ Equiv.Perm.cycleType h = Equiv.Perm.cycleType g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [this, ← Equiv.Perm.conj_class_card_mul_eq g] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Nat.factorial (Fintype.card α)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [Fintype.card_coe, ← Set.toFinset_card, mul_assoc] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [Equiv.permWithCycleType, Finset.mem_filter] at hg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [hg.2] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | apply congr_arg₂ _ _ rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card = (Set.toFinset {h | IsConj g h}).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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | apply congr_arg | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card = (Set.toFinset {h | IsConj g h}).card | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Finset.filter (fun h => IsConj g h) Finset.univ = Set.toFinset {h | IsConj g h} | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun h => IsConj g h) Finset.univ).card = (Set.toFinset {h | IsConj g h}).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | ext σ | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Finset.filter (fun h => IsConj g h) Finset.univ = Set.toFinset {h | IsConj g h} | case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
σ : Equiv.Perm α
⊢ σ ∈ Finset.filter (fun h => IsConj g h) Finset.univ ↔ σ ∈ Set.toFinset {h | IsConj g h} | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Finset.filter (fun h => IsConj g h) Finset.univ = Set.toFinset {h | IsConj g h}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [isConj_iff, Finset.mem_univ, forall_true_left, Finset.univ_filter_exists,
Finset.mem_image, true_and, Set.toFinset_setOf] | case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
σ : Equiv.Perm α
⊢ σ ∈ Finset.filter (fun h => IsConj g h) Finset.univ ↔ σ ∈ Set.toFinset {h | IsConj g h} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
this : Finset.filter (fun h => Equiv.Perm.cycleType h = m) Finset.univ = Finset.filter (fun h => IsConj g h) Finset.univ
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
σ : Equiv.Perm α
⊢ σ ∈ Finset.filter (fun h => IsConj g h) Finset.univ ↔ σ ∈ Set.toFinset {h | IsConj g h}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | convert MulZeroClass.zero_mul _ | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
0 | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0 | 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
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) =
0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [Finset.card_eq_zero] | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0 | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ = ∅ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [← Finset.isEmpty_coe_sort] | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ = ∅ | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ IsEmpty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ } | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ = ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [← not_nonempty_iff] | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ IsEmpty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ } | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ } | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ IsEmpty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | intro h | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ } | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | apply hm | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ False | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | simp only [Finset.nonempty_coe_sort] at h | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : Nonempty { x // x ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ }
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | rw [← Equiv.permWithCycleType_nonempty_iff] | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ (Equiv.permWithCycleType α m).Nonempty | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType_mul_eq | [3047, 1] | [3078, 12] | exact h | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ (Equiv.permWithCycleType α m).Nonempty | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
h : (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
⊢ (Equiv.permWithCycleType α m).Nonempty
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | split_ifs with hm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
if Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a then
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
else 0 | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0 | 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
m : Multiset ℕ
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
if Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a then
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
else 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | obtain ⟨g, hg⟩ := Equiv.permWithCycleType_nonempty_iff.mpr hm | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | simp only [Equiv.permWithCycleType, Finset.mem_filter] at hg | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Equiv.permWithCycleType α m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [← Equiv.Perm.conj_class_card_mul_eq g] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Nat.factorial (Fintype.card α) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | simp only [mul_assoc] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} * Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
Multiset.prod (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))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) * Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | simp only [Fintype.card_coe, ← Set.toFinset_card, mul_assoc] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
Fintype.card ↑{h | IsConj g h} *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [Nat.div_eq_of_eq_mul_left _] | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))) | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card =
(Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) /
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply congr_arg₂ | case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case pos.intro.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card = (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card
case pos.intro.hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) =
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card *
(Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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))))) =
(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card *
(Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m *
Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [hg.2] | case pos.intro.hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) =
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.hy
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum (Equiv.Perm.cycleType g)) *
(Multiset.prod (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)))) =
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply congr_arg | case pos.intro.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card = (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card | case pos.intro.hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Set.toFinset {h | IsConj g h} = Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.hx
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ (Set.toFinset {h | IsConj g h}).card = (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | ext h | case pos.intro.hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Set.toFinset {h | IsConj g h} = Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ h ∈ Set.toFinset {h | IsConj g h} ↔ h ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.hx.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Set.toFinset {h | IsConj g h} = Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | simp only [Set.toFinset_setOf, Finset.mem_univ, forall_true_left,
Finset.univ_filter_exists, Finset.mem_image, true_and, Finset.mem_filter] | case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ h ∈ Set.toFinset {h | IsConj g h} ↔ h ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ IsConj g h ↔ Equiv.Perm.cycleType h = m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ h ∈ Set.toFinset {h | IsConj g h} ↔ h ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [isConj_comm, Equiv.Perm.isConj_iff_cycleType_eq, hg.2] | case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ IsConj g h ↔ Equiv.Perm.cycleType h = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.hx.h.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
h : Equiv.Perm α
⊢ IsConj g h ↔ Equiv.Perm.cycleType h = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply Nat.mul_pos | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m))) | case ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum m) > 0
case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ 0 <
Nat.factorial (Fintype.card α - Multiset.sum m) *
(Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | exact Nat.factorial_pos (Fintype.card α - Multiset.sum m) | case ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum m) > 0
case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Nat.factorial (Fintype.card α - Multiset.sum m) > 0
case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply Nat.mul_pos | case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | case hb.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m > 0
case hb.hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m * Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply Multiset.prod_pos | case hb.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m > 0 | case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ m, 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod m > 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | intro a ha | case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ m, 0 < a | case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
ha : a ∈ m
⊢ 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ m, 0 < a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | exact lt_of_lt_of_le (by norm_num) (hm.2 a ha) | case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
ha : a ∈ m
⊢ 0 < a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.ha.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
ha : a ∈ m
⊢ 0 < a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | norm_num | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
ha : a ∈ m
⊢ 0 < 2 | 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
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
ha : a ∈ m
⊢ 0 < 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply Multiset.prod_pos | case hb.hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0 | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m), 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hb
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ Multiset.prod (Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m)) > 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | intro a | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m), 0 < a | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m) → 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
⊢ ∀ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m), 0 < a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [Multiset.mem_map] | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m) → 0 < a | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ (∃ a_1 ∈ Multiset.dedup m, Nat.factorial (Multiset.count a_1 m) = a) → 0 < a | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ a ∈ Multiset.map (fun n => Nat.factorial (Multiset.count n m)) (Multiset.dedup m) → 0 < a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rintro ⟨b, _, rfl⟩ | case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ (∃ a_1 ∈ Multiset.dedup m, Nat.factorial (Multiset.count a_1 m) = a) → 0 < a | case hb.hb.h.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
b : ℕ
left✝ : b ∈ Multiset.dedup m
⊢ 0 < Nat.factorial (Multiset.count b m) | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hb.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
a : ℕ
⊢ (∃ a_1 ∈ Multiset.dedup m, Nat.factorial (Multiset.count a_1 m) = a) → 0 < a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | apply Nat.factorial_pos | case hb.hb.h.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
b : ℕ
left✝ : b ∈ Multiset.dedup m
⊢ 0 < Nat.factorial (Multiset.count b m) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case hb.hb.h.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
g : Equiv.Perm α
hg : g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
b : ℕ
left✝ : b ∈ Multiset.dedup m
⊢ 0 < Nat.factorial (Multiset.count b m)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [Finset.card_eq_zero, ← Finset.not_nonempty_iff_eq_empty] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0 | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty | 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
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ (Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).card = 0
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | rw [← Equiv.permWithCycleType_nonempty_iff] at hm | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Equiv.permWithCycleType α m).Nonempty
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty | 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
m : Multiset ℕ
hm : ¬(Multiset.sum m ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Equiv.Perm.card_of_cycleType | [3083, 1] | [3120, 13] | exact hm | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬(Equiv.permWithCycleType α m).Nonempty
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty | 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
m : Multiset ℕ
hm : ¬(Equiv.permWithCycleType α m).Nonempty
⊢ ¬(Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ).Nonempty
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | split_ifs with hm | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
if Even (Multiset.sum m + Multiset.card m) then Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
else ∅ | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : ¬Even (Multiset.sum m + Multiset.card m)
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
∅ | 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
m : Multiset ℕ
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
if Even (Multiset.sum m + Multiset.card m) then Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
else ∅
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | ext g | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | 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
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
⊢ Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) =
Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | simp only [Finset.mem_image, Finset.mem_filter] | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.filter (fun g => Equiv.Perm.cycleType g = m) Finset.univ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | constructor | case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) →
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
case pos.a.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m →
g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) ↔
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | intro hg | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) →
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ)
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
⊢ g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ) →
g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | rw [Finset.mem_map] at hg | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ)
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
∃ a ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ,
{ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } a = g
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
g ∈
Finset.map { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ }
(Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ)
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | obtain ⟨⟨k, hk⟩, hk', rfl⟩ := hg | case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
∃ a ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ,
{ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } a = g
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m | case pos.a.mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
k : Equiv.Perm α
hk : k ∈ alternatingGroup α
hk' : { val := k, property := hk } ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ
⊢ { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk } ∈ Finset.univ ∧
Equiv.Perm.cycleType
({ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk }) =
m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g✝ : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
g : Equiv.Perm α
hg :
∃ a ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ,
{ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } a = g
⊢ g ∈ Finset.univ ∧ Equiv.Perm.cycleType g = m
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.AlternatingGroup.of_cycleType_eq | [3123, 1] | [3154, 19] | apply And.intro (Finset.mem_univ _) | case pos.a.mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
k : Equiv.Perm α
hk : k ∈ alternatingGroup α
hk' : { val := k, property := hk } ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ
⊢ { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk } ∈ Finset.univ ∧
Equiv.Perm.cycleType
({ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk }) =
m | case pos.a.mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
k : Equiv.Perm α
hk : k ∈ alternatingGroup α
hk' : { val := k, property := hk } ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ
⊢ Equiv.Perm.cycleType ({ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk }) =
m | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.a.mp.intro.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
m : Multiset ℕ
hm : Even (Multiset.sum m + Multiset.card m)
k : Equiv.Perm α
hk : k ∈ alternatingGroup α
hk' : { val := k, property := hk } ∈ Finset.filter (fun g => Equiv.Perm.cycleType ↑g = m) Finset.univ
⊢ { toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk } ∈ Finset.univ ∧
Equiv.Perm.cycleType
({ toFun := ⇑(Subgroup.subtype (alternatingGroup α)), inj' := ⋯ } { val := k, property := hk }) =
m
TACTIC:
|
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