internlm/Lean-Github huggingface dataset:
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2✝ : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
hk2 : orderOf k = 2
⊢ Nat.Prime 2
Tactic:
exact Nat.prime_two
Proof state after:
no goals
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case hg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
⊢ k ^ 2 = 1
Tactic:
ext x
Proof state after:
case hg.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
⊢ (k ^ 2) x = 1 x
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case hg.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
⊢ (k ^ 2) x = 1 x
Tactic:
simp only [Equiv.Perm.coe_one, id.def]
Proof state after:
case hg.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
⊢ (k ^ 2) x = x
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case hg.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
⊢ (k ^ 2) x = x
Tactic:
by_cases hx : x ∈ g.support
Proof state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∈ Equiv.Perm.support g
⊢ (k ^ 2) x = x

case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∉ Equiv.Perm.support g
⊢ (k ^ 2) x = x
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∈ Equiv.Perm.support g
⊢ (k ^ 2) x = x
Tactic:
obtain ⟨cx, hcx⟩ := Equiv.Perm.sameCycle_of_mem_support hx
Proof state after:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∈ Equiv.Perm.support g
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
⊢ (k ^ 2) x = x
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∈ Equiv.Perm.support g
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
⊢ (k ^ 2) x = x
Tactic:
obtain ⟨n, _, rfl⟩ := (hcx (a cx) (a.mem_support cx)).exists_pow_eq'
Proof state after:
case pos.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ (k ^ 2) ((g ^ n) (a cx)) = (g ^ n) (a cx)
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case pos.intro.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ (k ^ 2) ((g ^ n) (a cx)) = (g ^ n) (a cx)
Tactic:
convert hk_apply cx 2 n
Proof state after:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ cx = (τ ^ 2) cx
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ cx = (τ ^ 2) cx
Tactic:
suffices hτ2 : τ ^ 2 = 1 by
  rw [hτ2, Equiv.Perm.coe_one, id.def]
Proof state after:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ τ ^ 2 = 1
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ τ ^ 2 = 1
Tactic:
rw [pow_two]
Proof state after:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ τ * τ = 1
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case h.e'_3.h.e'_6.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
⊢ τ * τ = 1
Tactic:
rw [Equiv.swap_mul_self]
Proof state after:
no goals
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
n : ℕ
left✝ : n < orderOf g
hx : (g ^ n) (a cx) ∈ Equiv.Perm.support g
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y ((g ^ n) (a cx))
hτ2 : τ ^ 2 = 1
⊢ cx = (τ ^ 2) cx
Tactic:
rw [hτ2, Equiv.Perm.coe_one, id.def]
Proof state after:
no goals
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∉ Equiv.Perm.support g
⊢ (k ^ 2) x = x
Tactic:
rw [← Equiv.Perm.not_mem_support]
Proof state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
d : Equiv.Perm α
hd : d ∈ Equiv.Perm.cycleFactorsFinset g
hm : (Equiv.Perm.support c).card = (Equiv.Perm.support d).card
hm' : c ≠ d
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g } :=
  Equiv.swap { val := c, property := hc } { val := d, property := hd }
a : Equiv.Perm.Basis g
hτ : τ ∈ Iφ g
k : Equiv.Perm α := ConjAct.ofConjAct ↑((φ' a) { val := τ, property := hτ })
hk2 : ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), ConjAct.toConjAct k • ↑c = ↑(τ c)
hksup : Equiv.Perm.support k ≤ Equiv.Perm.support g
hk_apply :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
x : α
hx : x ∉ Equiv.Perm.support g
⊢ x ∉ Equiv.Perm.support (k ^ 2)