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.k_mul | [1775, 1] | [1791, 48] | rw [Equiv.Perm.coe_mul, Function.comp_apply, hσ, hτ] | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_mul | [1775, 1] | [1791, 48] | simp only [k_apply_of_not_mem_support x hx] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | ext x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
⊢ k a 1 = id | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
⊢ k a 1 x = id x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
⊢ k a 1 = id
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | by_cases hx : x ∈ g.support | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
⊢ k a 1 x = id x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a 1 x = id x
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a 1 x = id x | 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
a : Equiv.Perm.Basis g
x : α
⊢ k a 1 x = id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | simp only [id.def, k_apply_of_not_mem_support x hx] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a 1 x = id x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a 1 x = id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | simp only [mem_support_iff_exists_Kf a] at hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a 1 x = id x | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a 1 x = id x | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a 1 x = id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | obtain ⟨c, i, rfl⟩ := hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a 1 x = id x | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ k a 1 (Kf a 1 (c, i)) = id (Kf a 1 (c, i)) | 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
a : Equiv.Perm.Basis g
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a 1 x = id x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | rw [k_apply] | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ k a 1 (Kf a 1 (c, i)) = id (Kf a 1 (c, i)) | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a 1 (c, i) = id (Kf a 1 (c, i))
case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ k a 1 (Kf a 1 (c, i)) = id (Kf a 1 (c, i))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | rfl | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a 1 (c, i) = id (Kf a 1 (c, i))
case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a 1 (c, i) = id (Kf a 1 (c, i))
case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | intro c | case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_one | [1794, 1] | [1802, 54] | rfl | case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(1 c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [Fintype.bijective_iff_surjective_and_card] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Bijective (k a τ) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ) ∧ Fintype.card α = Fintype.card α | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Bijective (k a τ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | refine' And.intro _ rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ) ∧ Fintype.card α = Fintype.card α | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ) ∧ Fintype.card α = Fintype.card α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [Function.surjective_iff_hasRightInverse] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.HasRightInverse (k a τ) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.Surjective (k a τ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | use k a τ⁻¹ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.HasRightInverse (k a τ) | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.RightInverse (k a τ⁻¹) (k a τ) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.HasRightInverse (k a τ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [Function.rightInverse_iff_comp] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.RightInverse (k a τ⁻¹) (k a τ) | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a τ ∘ k a τ⁻¹ = id | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ Function.RightInverse (k a τ⁻¹) (k a τ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [k_mul] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a τ ∘ k a τ⁻¹ = id | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a (τ * τ⁻¹) = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a τ ∘ k a τ⁻¹ = id
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [mul_inv_self] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a (τ * τ⁻¹) = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a 1 = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a (τ * τ⁻¹) = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [k_one] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a 1 = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a 1 = id
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | exact hτ | case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hσ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | intro c | case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_bij | [1805, 1] | [1815, 56] | rw [← hτ (τ⁻¹ c), Equiv.Perm.apply_inv_self] | case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.hτ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ⁻¹ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | by_cases hx : x ∈ g.support | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | simp only [mem_support_iff_exists_Kf a] at hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | obtain ⟨d, j, rfl⟩ := hx | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j))) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | by_cases hcd : c = d | case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j))) | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j)))
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j))) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | rw [hcd, Equiv.Perm.Basis.Kf_apply',
k_apply a d _ hτ, k_apply a d _ hτ, ← Equiv.Perm.Basis.Kf_apply'] | case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j))) | case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ τ d = τ d
case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | rfl | case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ τ d = τ d
case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d | case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ τ d = τ d
case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | simp only [Equiv.Perm.coe_one, id.def] | case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.hd
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : c = d
⊢ d = 1 d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | rw [Equiv.Perm.Basis.Kf_apply'' a,
k_apply a _ _ hτ, Equiv.Perm.Basis.Kf_apply'' a] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j))) | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ τ c ≠ τ d
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ k a τ (↑c (Kf a 1 (d, j))) = ↑(τ c) (k a τ (Kf a 1 (d, j)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | exact (Equiv.injective τ).ne_iff.mpr hcd | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ τ c ≠ τ d
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ τ c ≠ τ d
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | exact hcd | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c d : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
j : ℤ
hcd : ¬c = d
⊢ c ≠ 1 d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | suffices ∀ (c : g.cycleFactorsFinset), (c : Equiv.Perm α) x = x by
simp only [this, k_apply_of_not_mem_support x hx] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ↑c x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | intro c | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ↑c x = x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ↑c x = x | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ↑c x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | rw [← Equiv.Perm.not_mem_support] | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ↑c x = x | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ x ∉ Equiv.Perm.support ↑c | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ↑c x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | apply Finset.not_mem_mono _ hx | case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ x ∉ Equiv.Perm.support ↑c | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ Equiv.Perm.support ↑c ⊆ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ x ∉ Equiv.Perm.support ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | exact Equiv.Perm.mem_cycleFactorsFinset_support_le c.prop | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ Equiv.Perm.support ↑c ⊆ Equiv.Perm.support g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ Equiv.Perm.support ↑c ⊆ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.k_cycle_apply | [1818, 1] | [1839, 62] | simp only [this, k_apply_of_not_mem_support x hx] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
this : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ↑c x = x
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
hx : x ∉ Equiv.Perm.support g
this : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ↑c x = x
⊢ k a τ (↑c x) = ↑(τ c) (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_eq_card_of_mem_range | [1842, 1] | [1847, 24] | obtain ⟨⟨k, hk⟩, rfl⟩ := hτ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hτ : τ ∈ MonoidHom.range (φ g)
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Equiv.Perm.support ↑(((φ g) { val := k, property := hk }) c)).card = (Equiv.Perm.support ↑c).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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hτ : τ ∈ MonoidHom.range (φ g)
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_eq_card_of_mem_range | [1842, 1] | [1847, 24] | rw [φ_eq, ConjAct.smul_def, Equiv.Perm.support_conj] | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Equiv.Perm.support ↑(((φ g) { val := k, property := hk }) c)).card = (Equiv.Perm.support ↑c).card | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Finset.map (Equiv.toEmbedding (ConjAct.ofConjAct k)) (Equiv.Perm.support ↑c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Equiv.Perm.support ↑(((φ g) { val := k, property := hk }) c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_eq_card_of_mem_range | [1842, 1] | [1847, 24] | apply Finset.card_map | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Finset.map (Equiv.toEmbedding (ConjAct.ofConjAct k)) (Equiv.Perm.support ↑c)).card = (Equiv.Perm.support ↑c).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
k : ConjAct (Equiv.Perm α)
hk : k ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ (Finset.map (Equiv.toEmbedding (ConjAct.ofConjAct k)) (Equiv.Perm.support ↑c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [MulAction.mem_stabilizer_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.toConjAct (φ'Fun a hτ) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.toConjAct (φ'Fun a hτ) • g = 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.toConjAct (φ'Fun a hτ) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [ConjAct.smul_def] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.toConjAct (φ'Fun a hτ) • g = g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)) * g * (ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)))⁻¹ = 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.toConjAct (φ'Fun a hτ) • g = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [ConjAct.ofConjAct_toConjAct] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)) * g * (ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)))⁻¹ = g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g * (φ'Fun a hτ)⁻¹ = 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)) * g * (ConjAct.ofConjAct (ConjAct.toConjAct (φ'Fun a hτ)))⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [mul_inv_eq_iff_eq_mul] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g * (φ'Fun a hτ)⁻¹ = g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g = g * φ'Fun a hτ | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g * (φ'Fun a hτ)⁻¹ = g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | ext x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g = g * φ'Fun a hτ | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (φ'Fun a hτ * g) x = (g * φ'Fun a hτ) x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ φ'Fun a hτ * g = g * φ'Fun a hτ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | simp only [Equiv.Perm.coe_mul] | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (φ'Fun a hτ * g) x = (g * φ'Fun a hτ) x | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(φ'Fun a hτ) ∘ ⇑g) x = (⇑g ∘ ⇑(φ'Fun a hτ)) x | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (φ'Fun a hτ * g) x = (g * φ'Fun a hτ) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | simp only [φ'Fun] | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(φ'Fun a hτ) ∘ ⇑g) x = (⇑g ∘ ⇑(φ'Fun a hτ)) x | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(Equiv.ofBijective (k a τ) ⋯) ∘ ⇑g) x = (⇑g ∘ ⇑(Equiv.ofBijective (k a τ) ⋯)) x | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(φ'Fun a hτ) ∘ ⇑g) x = (⇑g ∘ ⇑(φ'Fun a hτ)) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | simp only [Function.comp_apply, Equiv.ofBijective_apply] | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(Equiv.ofBijective (k a τ) ⋯) ∘ ⇑g) x = (⇑g ∘ ⇑(Equiv.ofBijective (k a τ) ⋯)) x | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ k a τ (g x) = g (k a τ x) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑(Equiv.ofBijective (k a τ) ⋯) ∘ ⇑g) x = (⇑g ∘ ⇑(Equiv.ofBijective (k a τ) ⋯)) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [← Function.comp_apply (f := k a τ)] | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ k a τ (g x) = g (k a τ x) | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (k a τ ∘ ⇑g) x = g (k a τ x) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ k a τ (g x) = g (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rw [k_commute hτ] | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (k a τ ∘ ⇑g) x = g (k a τ x) | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑g ∘ k a τ) x = g (k a τ x) | 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
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (k a τ ∘ ⇑g) x = g (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_mem_stabilizer | [1857, 1] | [1871, 6] | rfl | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑g ∘ k a τ) x = g (k a τ x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (⇑g ∘ k a τ) x = g (k a τ x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.mem_Iφ_iff | [1900, 1] | [1902, 22] | simp only [Iφ] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g ↔
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈
{
toSubmonoid :=
{
toSubsemigroup :=
{
carrier :=
{τ |
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card},
mul_mem' := ⋯ },
one_mem' := ⋯ },
inv_mem' := ⋯ } ↔
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).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
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g ↔
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.mem_Iφ_iff | [1900, 1] | [1902, 22] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈
{
toSubmonoid :=
{
toSubsemigroup :=
{
carrier :=
{τ |
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card},
mul_mem' := ⋯ },
one_mem' := ⋯ },
inv_mem' := ⋯ } ↔
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | 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
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈
{
toSubmonoid :=
{
toSubsemigroup :=
{
carrier :=
{τ |
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card},
mul_mem' := ⋯ },
one_mem' := ⋯ },
inv_mem' := ⋯ } ↔
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | intro x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ Equiv.Perm.support (ConjAct.ofConjAct ↑((φ' a) τ)) ≤ Equiv.Perm.support g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ x ∈ Equiv.Perm.support (ConjAct.ofConjAct ↑((φ' a) τ)) → x ∈ Equiv.Perm.support g | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ Equiv.Perm.support (ConjAct.ofConjAct ↑((φ' a) τ)) ≤ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | simp only [Equiv.Perm.mem_support] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ x ∈ Equiv.Perm.support (ConjAct.ofConjAct ↑((φ' a) τ)) → x ∈ Equiv.Perm.support g | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x → g x ≠ x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ x ∈ Equiv.Perm.support (ConjAct.ofConjAct ↑((φ' a) τ)) → x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | intro hx' hx | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x → g x ≠ x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ False | 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
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x → g x ≠ x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | apply hx' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ False | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | rw [← Equiv.Perm.not_mem_support] at hx | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : x ∉ Equiv.Perm.support g
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : g x = x
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.φ'_support_le | [1954, 1] | [1962, 55] | exact OnCycleFactors.k_apply_of_not_mem_support x hx | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : x ∉ Equiv.Perm.support g
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
x : α
hx' : (ConjAct.ofConjAct ↑((φ' a) τ)) x ≠ x
hx : x ∉ Equiv.Perm.support g
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) x = x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_equivariant | [1965, 1] | [1970, 48] | rw [ConjAct.smul_def] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ↑((φ' a) τ) • ↑c = ↑(↑τ c) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹ = ↑(↑τ c) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ↑((φ' a) τ) • ↑c = ↑(↑τ c)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_equivariant | [1965, 1] | [1970, 48] | rw [mul_inv_eq_iff_eq_mul] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹ = ↑(↑τ c) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c = ↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ) | 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
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹ = ↑(↑τ c)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_equivariant | [1965, 1] | [1970, 48] | ext x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c = ↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ) | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ) * ↑c) x = (↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ)) x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ ConjAct.ofConjAct ↑((φ' a) τ) * ↑c = ↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_equivariant | [1965, 1] | [1970, 48] | exact OnCycleFactors.k_cycle_apply τ.prop c x | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ) * ↑c) x = (↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ)) x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ) * ↑c) x = (↑(↑τ c) * ConjAct.ofConjAct ↑((φ' a) τ)) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | apply Equiv.Perm.ext | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ (φ g) ((φ' a) τ) = ↑τ | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), ((φ g) ((φ' a) τ)) x = ↑τ x | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ (φ g) ((φ' a) τ) = ↑τ
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | rintro ⟨c, hc⟩ | case H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), ((φ g) ((φ' a) τ)) x = ↑τ x | case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ((φ g) ((φ' a) τ)) { val := c, property := hc } = ↑τ { val := c, property := hc } | 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
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), ((φ g) ((φ' a) τ)) x = ↑τ x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | rw [← Subtype.coe_inj] | case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ((φ g) ((φ' a) τ)) { val := c, property := hc } = ↑τ { val := c, property := hc } | case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(((φ g) ((φ' a) τ)) { val := c, property := hc }) = ↑(↑τ { val := c, property := hc }) | Please generate a tactic in lean4 to solve the state.
STATE:
case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ((φ g) ((φ' a) τ)) { val := c, property := hc } = ↑τ { val := c, property := hc }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | convert φ_eq'2 g ((φ' a τ)) ⟨c, hc⟩ | case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(((φ g) ((φ' a) τ)) { val := c, property := hc }) = ↑(↑τ { val := c, property := hc }) | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc } * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case H.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(((φ g) ((φ' a) τ)) { val := c, property := hc }) = ↑(↑τ { val := c, property := hc })
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | rw [eq_mul_inv_iff_mul_eq] | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc } * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹ | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc } | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc } * (ConjAct.ofConjAct ↑((φ' a) τ))⁻¹
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | ext x | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc } | case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ)) x =
(ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc }) x | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ) =
ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | simp only [Finset.coe_sort_coe, Equiv.Perm.coe_mul, Function.comp_apply] | case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ)) x =
(ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc }) x | case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x) = (ConjAct.ofConjAct ↑((φ' a) τ)) (c x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (↑(↑τ { val := c, property := hc }) * ConjAct.ofConjAct ↑((φ' a) τ)) x =
(ConjAct.ofConjAct ↑((φ' a) τ) * ↑{ val := c, property := hc }) x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | apply symm | case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x) = (ConjAct.ofConjAct ↑((φ' a) τ)) (c x) | case h.e'_3.H.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) (c x) = ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x) = (ConjAct.ofConjAct ↑((φ' a) τ)) (c x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ'_is_rightInverse | [1977, 1] | [1987, 41] | exact (k_cycle_apply τ.prop ⟨c, hc⟩ x) | case h.e'_3.H.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) (c x) = ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.H.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : ↥(Iφ g)
c : Equiv.Perm α
hc : c ∈ ↑(Equiv.Perm.cycleFactorsFinset g)
x : α
⊢ (ConjAct.ofConjAct ↑((φ' a) τ)) (c x) = ↑(↑τ { val := c, property := hc }) ((ConjAct.ofConjAct ↑((φ' a) τ)) x)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | obtain ⟨a⟩ := g.existsBasis | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Iφ g = MonoidHom.range (φ g) | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
⊢ Iφ g = MonoidHom.range (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Iφ g = MonoidHom.range (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | ext τ | case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
⊢ Iφ g = MonoidHom.range (φ g) | case intro.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g ↔ τ ∈ MonoidHom.range (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
⊢ Iφ g = MonoidHom.range (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | constructor | case intro.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g ↔ τ ∈ MonoidHom.range (φ g) | case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g → τ ∈ MonoidHom.range (φ g)
case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) → τ ∈ Iφ g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g ↔ τ ∈ MonoidHom.range (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | intro hτ | case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g → τ ∈ MonoidHom.range (φ g) | case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ : τ ∈ Iφ g
⊢ τ ∈ MonoidHom.range (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ Iφ g → τ ∈ MonoidHom.range (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | exact ⟨(φ' a) ⟨τ, hτ⟩, hφ'_is_rightInverse ⟨τ, hτ⟩⟩ | case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ : τ ∈ Iφ g
⊢ τ ∈ MonoidHom.range (φ g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h.mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ : τ ∈ Iφ g
⊢ τ ∈ MonoidHom.range (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | rw [mem_Iφ_iff] | case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) → τ ∈ Iφ g | case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) →
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) → τ ∈ Iφ g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.Iφ_eq_range | [1990, 1] | [1997, 34] | exact hφ_eq_card_of_mem_range | case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) →
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.h.mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ τ ∈ MonoidHom.range (φ g) →
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_range_iff | [2001, 1] | [2005, 6] | simp only [← Iφ_eq_range, mem_Iφ_iff] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ MonoidHom.range (φ g) ↔
∀ (c : ↑↑(Equiv.Perm.cycleFactorsFinset g)), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (c : ↑↑(Equiv.Perm.cycleFactorsFinset g)), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ MonoidHom.range (φ g) ↔
∀ (c : ↑↑(Equiv.Perm.cycleFactorsFinset g)), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_mem_range_iff | [2001, 1] | [2005, 6] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (c : ↑↑(Equiv.Perm.cycleFactorsFinset g)), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card | 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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (c : ↑↑(Equiv.Perm.cycleFactorsFinset g)), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | rw [← Iφ_eq_range] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ↑(MonoidHom.range (φ g)) = {τ | fsc ∘ ⇑τ = fsc} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ↑(Iφ g) = {τ | fsc ∘ ⇑τ = fsc} | 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
⊢ ↑(MonoidHom.range (φ g)) = {τ | fsc ∘ ⇑τ = fsc}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | ext τ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ↑(Iφ g) = {τ | fsc ∘ ⇑τ = fsc} | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ τ ∈ {τ | fsc ∘ ⇑τ = fsc} | 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
⊢ ↑(Iφ g) = {τ | fsc ∘ ⇑τ = fsc}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | simp only [Finset.coe_sort_coe, Set.mem_setOf_eq, Function.funext_iff, Function.comp_apply] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ τ ∈ {τ | fsc ∘ ⇑τ = fsc} | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a | 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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ τ ∈ {τ | fsc ∘ ⇑τ = fsc}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | simp only [SetLike.mem_coe, mem_Iφ_iff] | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a | 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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ τ ∈ ↑(Iφ g) ↔ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | apply forall_congr' | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ a)).card = (Equiv.Perm.support ↑a).card ↔ fsc (τ a) = fsc a | 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
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ (∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card) ↔
∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), fsc (τ a) = fsc a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | intro c | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ a)).card = (Equiv.Perm.support ↑a).card ↔ fsc (τ a) = fsc a | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card ↔ fsc (τ c) = fsc c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ∀ (a : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
(Equiv.Perm.support ↑(τ a)).card = (Equiv.Perm.support ↑a).card ↔ fsc (τ a) = fsc a
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range' | [2020, 1] | [2029, 65] | simp only [← Function.Injective.eq_iff Fin.val_injective, fsc] | case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card ↔ fsc (τ c) = fsc c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
τ : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card ↔ fsc (τ c) = fsc c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card' | [2032, 1] | [2036, 6] | simp_rw [← hφ_range', Nat.card_eq_fintype_card] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Nat.card ↥(MonoidHom.range (φ g)) = Fintype.card ↑{k | fsc ∘ ⇑k = fsc} | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) = Fintype.card ↑↑(MonoidHom.range (φ g)) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Nat.card ↥(MonoidHom.range (φ g)) = Fintype.card ↑{k | fsc ∘ ⇑k = fsc}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card' | [2032, 1] | [2036, 6] | rfl | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) = Fintype.card ↑↑(MonoidHom.range (φ g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Fintype.card ↥(MonoidHom.range (φ g)) = Fintype.card ↑↑(MonoidHom.range (φ g))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [hφ_range_card', Equiv.Perm.of_partition_card] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ Nat.card ↥(MonoidHom.range (φ 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
⊢ (Finset.prod Finset.univ fun i => Nat.factorial (Fintype.card ↑{a | fsc a = i})) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g))) | 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
⊢ Nat.card ↥(MonoidHom.range (φ g)) =
Multiset.prod
(Multiset.map (fun n => Nat.factorial (Multiset.count n (Equiv.Perm.cycleType g)))
(Multiset.dedup (Equiv.Perm.cycleType g)))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rintro ⟨i, hi⟩ | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g) | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi }) (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
⊢ ∀ (n : Fin (Fintype.card α + 1)), Fintype.card ↑{a | fsc a = n} = Multiset.count (↑n) (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [Equiv.Perm.cycleType_def] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi }) (Equiv.Perm.cycleType g) | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi })
(Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi }) (Equiv.Perm.cycleType g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Fin.val_mk] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi })
(Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count i (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count (↑{ val := i, isLt := hi })
(Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [← Set.ncard_filter_eq_count (Finset.card ∘ Equiv.Perm.support) g.cycleFactorsFinset i] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count i (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val) | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Multiset.count i (Multiset.map (Finset.card ∘ Equiv.Perm.support) (Equiv.Perm.cycleFactorsFinset g).val)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | simp only [Set.coe_setOf, Set.mem_setOf_eq, Function.comp_apply] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card ↑{a | fsc a = { val := i, isLt := hi }} =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | let u : {x : g.cycleFactorsFinset | fsc x = ⟨i, hi⟩} →
{x ∈ g.cycleFactorsFinset | (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun ⟨⟨x, hx⟩, hx'⟩ => ⟨x, by
simp only [fsc, Fin.mk.injEq, Set.mem_setOf_eq] at hx'
simp only [Function.comp_apply, Set.mem_setOf_eq]
exact ⟨hx, hx'⟩⟩ | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | rw [← Set.Nat.card_coe_set_eq, Nat.card_eq_fintype_card] | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Fintype.card ↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Set.ncard {x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i}
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.hφ_range_card | [2039, 1] | [2088, 39] | apply Fintype.card_of_bijective this | case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Fintype.card ↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
i : ℕ
hi : i < Fintype.card α + 1
u : ↑{x | fsc x = { val := i, isLt := hi }} →
↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Finset.card ∘ Equiv.Perm.support) x = i} :=
fun x =>
match x with
| { val := { val := x, property := hx }, property := hx' } => { val := x, property := ⋯ }
this : Function.Bijective u
⊢ Fintype.card { x // fsc x = { val := i, isLt := hi } } =
Fintype.card ↑{x | x ∈ Equiv.Perm.cycleFactorsFinset g ∧ (Equiv.Perm.support x).card = i}
TACTIC:
|
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