internlm/Lean-Github huggingface dataset:
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : (g ^ n) (a (τ cx)) ≠ (g ^ n) (a cx)
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
⊢ c = ↑cx ∨ d = ↑cx
Tactic:
rw [Function.Injective.ne_iff (Equiv.injective _)] at hx
Proof state after:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
⊢ c = ↑cx ∨ d = ↑cx
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
⊢ c = ↑cx ∨ d = ↑cx
Tactic:
have hx' : τ cx ≠ cx := ne_of_apply_ne a hx
Proof state after:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : τ cx ≠ cx
⊢ c = ↑cx ∨ d = ↑cx
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : τ cx ≠ cx
⊢ c = ↑cx ∨ d = ↑cx
Tactic:
rw [← Equiv.Perm.mem_support] at hx'
Proof state after:
case a.mp.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
⊢ c = ↑cx ∨ d = ↑cx
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd }
Tactic:
intro h
Proof state after:
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : { val := c, property := hc } = { val := d, property := hd }
⊢ False
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : { val := c, property := hc } = { val := d, property := hd }
⊢ False
Tactic:
rw [← Subtype.coe_inj] at h
Proof state after:
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : ↑{ val := c, property := hc } = ↑{ val := d, property := hd }
⊢ False
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : ↑{ val := c, property := hc } = ↑{ val := d, property := hd }
⊢ False
Tactic:
apply hm'
Proof state after:
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : ↑{ val := c, property := hc } = ↑{ val := d, property := hd }
⊢ c = d
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx ∈ Equiv.Perm.support τ
h : ↑{ val := c, property := hc } = ↑{ val := d, property := hd }
⊢ c = d
Tactic:
exact h
Proof state after:
no goals
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.intro.intro.intro.inl
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx = { val := c, property := hc }
⊢ c = ↑cx ∨ d = ↑cx
Tactic:
apply Or.intro_left
Proof state after:
case a.mp.intro.intro.intro.inl.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx = { val := c, property := hc }
⊢ c = ↑cx
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.intro.intro.intro.inl.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx = { val := c, property := hc }
⊢ c = ↑cx
Tactic:
rw [hx']
Proof state after:
no goals
Theorem: OnCycleFactors.count_le_one_of_mem_kerφ
Proof state before:
case a.mp.intro.intro.intro.inr
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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx = { val := d, property := hd }
⊢ c = ↑cx ∨ d = ↑cx
Tactic:
apply Or.intro_right
Proof state after:
case a.mp.intro.intro.intro.inr.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
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
this' : Multiset.card (Equiv.Perm.cycleType k) * 2 = (Equiv.Perm.support k).card
x : α
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hxcx : x ∈ Equiv.Perm.support ↑cx
n : ℕ
hx : a (τ cx) ≠ a cx
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
hx' : cx = { val := d, property := hd }
⊢ d = ↑cx