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.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hx' | case 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
hx : x ∈ Equiv.Perm.support c ∨ x ∈ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : ¬(cx = { val := c, property := hc } ∨ cx = { val := d, property := hd })
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd } | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd } | Please generate a tactic in lean4 to solve the state.
STATE:
case 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
hx : x ∈ Equiv.Perm.support c ∨ x ∈ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : ¬(cx = { val := c, property := hc } ∨ cx = { val := d, property := hd })
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [Equiv.Perm.disjoint_iff_support_disjoint] | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx) | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ¬Disjoint (Equiv.Perm.support ↑cx) (Equiv.Perm.support ↑(τ cx)) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [Finset.not_disjoint_iff] | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ¬Disjoint (Equiv.Perm.support ↑cx) (Equiv.Perm.support ↑(τ cx)) | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ∃ a ∈ Equiv.Perm.support ↑cx, a ∈ Equiv.Perm.support ↑(τ cx) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ¬Disjoint (Equiv.Perm.support ↑cx) (Equiv.Perm.support ↑(τ cx))
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | use a cx | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ∃ a ∈ Equiv.Perm.support ↑cx, a ∈ Equiv.Perm.support ↑(τ cx) | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑cx ∧ a cx ∈ Equiv.Perm.support ↑(τ cx) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ ∃ a ∈ Equiv.Perm.support ↑cx, a ∈ Equiv.Perm.support ↑(τ cx)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply And.intro (a.mem_support cx) | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑cx ∧ a cx ∈ Equiv.Perm.support ↑(τ cx) | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑(τ cx) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑cx ∧ a cx ∈ Equiv.Perm.support ↑(τ cx)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [← haτcx_eq_acx] | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑(τ cx) | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a (τ cx) ∈ Equiv.Perm.support ↑(τ cx) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a cx ∈ Equiv.Perm.support ↑(τ cx)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact a.mem_support (τ cx) | case 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a (τ cx) ∈ Equiv.Perm.support ↑(τ cx) | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
⊢ a (τ cx) ∈ Equiv.Perm.support ↑(τ cx)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | intro 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd } | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : { val := c, property := hc } = { val := d, property := hd }
⊢ 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
⊢ { val := c, property := hc } ≠ { val := d, property := hd }
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | simp only [← Subtype.coe_inj, Subtype.coe_mk] at 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : { val := c, property := hc } = { val := d, property := hd }
⊢ False | 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : c = d
⊢ 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : { val := c, property := hc } = { val := d, property := hd }
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hm' 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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : c = d
⊢ False | 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
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
cx : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hcx : ∀ y ∈ Equiv.Perm.support ↑cx, Equiv.Perm.SameCycle g y x
hcx' : ↑cx = Equiv.Perm.cycleOf g x
n : ℕ
left✝ : n < orderOf g
hnx : (g ^ n) (a cx) = x
hk_apply : ∀ (n : ℕ), k ((g ^ n) (a cx)) = (g ^ n) (a (τ cx))
haτcx_eq_acx : a (τ cx) = a cx
this : ¬Equiv.Perm.Disjoint ↑cx ↑(τ cx)
this' : cx ∉ Equiv.Perm.support τ
h : c = d
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [← Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff] | case a.mpr.refine_2
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
⊢ x ∈ Equiv.Perm.support g | case a.mpr.refine_2
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
⊢ x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | cases' hx' with hxc hxd | case a.mpr.refine_2
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hx' : Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hxc] | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ c ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hc | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ c ∈ Equiv.Perm.cycleFactorsFinset g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxc : Equiv.Perm.cycleOf g x = c
⊢ c ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hxd] | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ d ∈ Equiv.Perm.cycleFactorsFinset g | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ Equiv.Perm.cycleOf g x ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hd | case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ d ∈ Equiv.Perm.cycleFactorsFinset g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_2.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
hxd : Equiv.Perm.cycleOf g x = d
⊢ d ∈ Equiv.Perm.cycleFactorsFinset g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | simp only [Finset.mem_union] at hx | case a.mpr.refine_1
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d | case a.mpr.refine_1
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))
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
hx : x ∈ Equiv.Perm.support c ∨ x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1
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))
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
hx : x ∈ Equiv.Perm.support c ∪ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | cases hx with
| inl hx =>
apply Or.intro_left
exact (Equiv.Perm.cycle_is_cycleOf hx hc).symm
| inr hx =>
apply Or.intro_right
exact (Equiv.Perm.cycle_is_cycleOf hx hd).symm | case a.mpr.refine_1
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))
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
hx : x ∈ Equiv.Perm.support c ∨ x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1
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))
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
hx : x ∈ Equiv.Perm.support c ∨ x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply Or.intro_left | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = c | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact (Equiv.Perm.cycle_is_cycleOf hx hc).symm | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support c
⊢ Equiv.Perm.cycleOf g x = c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply Or.intro_right | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = d | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = c ∨ Equiv.Perm.cycleOf g x = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact (Equiv.Perm.cycle_is_cycleOf hx hd).symm | case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr.refine_1.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
hk_apply :
∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) (m n : ℕ), (k ^ m) ((g ^ n) (a c)) = (g ^ n) (a ((τ ^ m) c))
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
hx : x ∈ Equiv.Perm.support d
⊢ Equiv.Perm.cycleOf g x = d
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply Equiv.Perm.sum_cycleType | case e_n
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))
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
⊢ Multiset.sum (Equiv.Perm.cycleType k) = (Equiv.Perm.support k).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_n
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))
this : Equiv.Perm.cycleType k = Multiset.replicate (Multiset.card (Equiv.Perm.cycleType k)) 2
⊢ Multiset.sum (Equiv.Perm.cycleType k) = (Equiv.Perm.support k).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | have hk2' : Nat.Prime (orderOf k) := by
rw [hk2]
exact Nat.prime_two | 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
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2 | 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
hk2' : Nat.Prime (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2 | 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
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
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | obtain ⟨n, hn⟩ := Equiv.Perm.cycleType_prime_order hk2' | 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
hk2' : Nat.Prime (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2 | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2 | 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
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
hk2' : Nat.Prime (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | intro i hi | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2 | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType k
⊢ i = 2 | 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
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
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
⊢ ∀ i ∈ Equiv.Perm.cycleType k, i = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hn, hk2, Multiset.mem_replicate] at hi | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType k
⊢ i = 2 | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : n + 1 ≠ 0 ∧ i = 2
⊢ i = 2 | 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
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
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : i ∈ Equiv.Perm.cycleType k
⊢ i = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hi.right | case 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))
hk2 : orderOf k = 2
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : n + 1 ≠ 0 ∧ i = 2
⊢ i = 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
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
hk2' : Nat.Prime (orderOf k)
n : ℕ
hn : Equiv.Perm.cycleType k = Multiset.replicate (n + 1) (orderOf k)
i : ℕ
hi : n + 1 ≠ 0 ∧ i = 2
⊢ i = 2
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hk2] | 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 (orderOf k) | 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 | 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
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 (orderOf k)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact Nat.prime_two | 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 | 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
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | ext x | 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 | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | simp only [Equiv.Perm.coe_one, id.def] | 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 | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | by_cases hx : x ∈ g.support | 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 | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | obtain ⟨cx, hcx⟩ := Equiv.Perm.sameCycle_of_mem_support hx | 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 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 | 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
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | obtain ⟨n, _, rfl⟩ := (hcx (a cx) (a.mem_support cx)).exists_pow_eq' | 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 | 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) | Please generate a tactic in lean4 to solve the state.
STATE:
case pos.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | convert hk_apply cx 2 n | 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) | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | suffices hτ2 : τ ^ 2 = 1 by
rw [hτ2, Equiv.Perm.coe_one, id.def] | 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 | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [pow_two] | 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 | 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 | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [Equiv.swap_mul_self] | 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 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hτ2, Equiv.Perm.coe_one, id.def] | 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 | 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
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [← Equiv.Perm.not_mem_support] | 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 | 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) | 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
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:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | intro hx' | 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) | 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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ False | 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
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)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply hx | 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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ False | 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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ 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
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ False
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply hksup | 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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support g | case neg.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support k | 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
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | apply Equiv.Perm.support_pow_le k 2 | case neg.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support k | case neg.a.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support (k ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support k
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hx' | case neg.a.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support (k ^ 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.a.a
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
hx' : x ∈ Equiv.Perm.support (k ^ 2)
⊢ x ∈ Equiv.Perm.support (k ^ 2)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | intro x | case intro.intro.intro.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
⊢ τ ∈ Iφ g | case intro.intro.intro.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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.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
⊢ τ ∈ Iφ g
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | by_cases hx : x = ⟨c, hc⟩ | case intro.intro.intro.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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | by_cases hx' : x = ⟨d, hd⟩ | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : ¬x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hx, Equiv.swap_apply_left] | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑{ val := d, property := hd }).card = (Equiv.Perm.support ↑{ val := c, property := hc }).card | 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
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hm.symm | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑{ val := d, property := hd }).card = (Equiv.Perm.support ↑{ val := c, property := hc }).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : x = { val := c, property := hc }
⊢ (Equiv.Perm.support ↑{ val := d, property := hd }).card = (Equiv.Perm.support ↑{ val := c, property := hc }).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [hx', Equiv.swap_apply_right] | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑{ val := c, property := hc }).card = (Equiv.Perm.support ↑{ val := d, property := hd }).card | 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
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | exact hm | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑{ val := c, property := hc }).card = (Equiv.Perm.support ↑{ val := d, property := hd }).card | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑{ val := c, property := hc }).card = (Equiv.Perm.support ↑{ val := d, property := hd }).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.count_le_one_of_mem_kerφ | [3356, 1] | [3558, 47] | rw [Equiv.swap_apply_of_ne_of_ne hx hx'] | 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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : ¬x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card | 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
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
x : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hx : ¬x = { val := c, property := hc }
hx' : ¬x = { val := d, property := hd }
⊢ (Equiv.Perm.support ↑(τ x)).card = (Equiv.Perm.support ↑x).card
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [SetLike.le_def] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α ↔
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) ↔
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g ≤ alternatingGroup α ↔
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | constructor | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) ↔
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) →
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ((∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1) →
∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup 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
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) ↔
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | intro h | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) →
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ (∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ (∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α) →
(∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | constructor | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ (∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ i ∈ Equiv.Perm.cycleType g, Odd i
case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ (∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact odd_of_mem_kerφ h | case mp.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ i ∈ Equiv.Perm.cycleType g, Odd i
case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ i ∈ Equiv.Perm.cycleType g, Odd i
case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | constructor | case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp.right.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact card_le_of_mem_kerφ h | case mp.right.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact count_le_one_of_mem_kerφ h | case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.right.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h : ∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
⊢ ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rintro ⟨h_odd, h_fixed, h_count⟩ x hx | case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ((∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1) →
∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ x ∈ alternatingGroup α | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
⊢ ((∀ i ∈ Equiv.Perm.cycleType g, Odd i) ∧
Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 ∧
∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1) →
∀ ⦃x : ConjAct (Equiv.Perm α)⦄, x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g → x ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | suffices (OnCycleFactors.φ g).ker = ⊤ by
rw [← OnCycleFactors.hφ_ker_eq_θ_range, this]
simp only [Subgroup.coeSubtype, Subgroup.mem_map, Subgroup.mem_top, true_and]
exact ⟨⟨x, hx⟩, rfl⟩ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ x ∈ Set.range ⇑(θ g) | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ MonoidHom.ker (φ g) = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ x ∈ Set.range ⇑(θ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [eq_top_iff] | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ MonoidHom.ker (φ g) = ⊤ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ ⊤ ≤ MonoidHom.ker (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ MonoidHom.ker (φ g) = ⊤
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | intro y _ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ ⊤ ≤ MonoidHom.ker (φ g) | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ y ∈ MonoidHom.ker (φ g) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
⊢ ⊤ ≤ MonoidHom.ker (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | suffices (OnCycleFactors.φ g).range = ⊥ by
rw [MonoidHom.mem_ker, ← Subgroup.mem_bot, ← this, MonoidHom.mem_range]
exact ⟨y, rfl⟩ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ y ∈ MonoidHom.ker (φ g) | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) = ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ y ∈ MonoidHom.ker (φ g)
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [eq_bot_iff] | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) = ⊥ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) ≤ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) = ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | intro z | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) ≤ ⊥ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ MonoidHom.range (φ g) → z ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
⊢ MonoidHom.range (φ g) ≤ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [← OnCycleFactors.Iφ_eq_range, Subgroup.mem_bot] | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ MonoidHom.range (φ g) → z ∈ ⊥ | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ Iφ g → z = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ MonoidHom.range (φ g) → z ∈ ⊥
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | intro hz | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ Iφ g → z = 1 | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ z = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z ∈ Iφ g → z = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | apply Equiv.ext | case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ z = 1 | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), z x = 1 x | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ z = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | intro c | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), z x = 1 x | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
⊢ ∀ (x : ↑↑(Equiv.Perm.cycleFactorsFinset g)), z x = 1 x
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [← Multiset.nodup_iff_count_le_one, Equiv.Perm.cycleType_def,
Multiset.nodup_map_iff_inj_on (Equiv.Perm.cycleFactorsFinset g).nodup]
at h_count | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [Equiv.Perm.coe_one, id.def, ← Subtype.coe_inj] | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(z c) = ↑c | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ z c = 1 c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact h_count (z c) (z c).prop c c.prop (hz c) | case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(z c) = ↑c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro.H
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count :
∀ x ∈ (Equiv.Perm.cycleFactorsFinset g).val,
∀ y ∈ (Equiv.Perm.cycleFactorsFinset g).val,
(Finset.card ∘ Equiv.Perm.support) x = (Finset.card ∘ Equiv.Perm.support) y → x = y
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
y : ↥(MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)
a✝ : y ∈ ⊤
z : Equiv.Perm ↑↑(Equiv.Perm.cycleFactorsFinset g)
hz : z ∈ Iφ g
c : ↑↑(Equiv.Perm.cycleFactorsFinset g)
⊢ ↑(z c) = ↑c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | obtain ⟨⟨y, uv⟩, rfl⟩ := Set.mem_range.mp hx' | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : x ∈ Set.range ⇑(θ g)
⊢ x ∈ alternatingGroup α | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (θ g) (y, uv) ∈ alternatingGroup 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
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : x ∈ Set.range ⇑(θ g)
⊢ x ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [Equiv.Perm.mem_alternatingGroup, OnCycleFactors.sign_ψ] | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (θ g) (y, uv) ∈ alternatingGroup α | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.sign y * Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1 | 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
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (θ g) (y, uv) ∈ alternatingGroup α
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | convert mul_one _ | case intro.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.sign y * Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1
case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ 1 = Equiv.Perm.sign y | 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
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.sign y * Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | apply Finset.prod_eq_one | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1 | case h.e'_2.h.e'_6.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ ∀ x ∈ Finset.univ, Equiv.Perm.sign ↑(uv x) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Finset.prod Finset.univ fun c => Equiv.Perm.sign ↑(uv c)) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rintro ⟨c, hc⟩ _ | case h.e'_2.h.e'_6.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ ∀ x ∈ Finset.univ, Equiv.Perm.sign ↑(uv x) = 1 | case h.e'_2.h.e'_6.h.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ ∀ x ∈ Finset.univ, Equiv.Perm.sign ↑(uv x) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | obtain ⟨k, hk⟩ := (uv _).prop | case h.e'_2.h.e'_6.h.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1 | case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑?m.575606 ^ x) k = ↑(uv ?m.575606)
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.h.mk
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [← hk, map_zpow] | case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑?m.575606 ^ x) k = ↑(uv ?m.575606)
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1 | case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } ^ k = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑?m.575606 ^ x) k = ↑(uv ?m.575606)
⊢ Equiv.Perm.sign ↑(uv { val := c, property := hc }) = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | convert one_zpow k | case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } ^ k = 1 | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6.h.mk.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } ^ k = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | simp only | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } = 1 | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign c = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign ↑{ val := c, property := hc } = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [Equiv.Perm.IsCycle.sign, Odd.neg_one_pow, neg_neg] | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign c = 1 | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Odd (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.sign c = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | apply h_odd | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Odd (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ (Equiv.Perm.support c).card ∈ Equiv.Perm.cycleType g
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Odd (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [Equiv.Perm.cycleType_def, Multiset.mem_map] | case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ (Equiv.Perm.support c).card ∈ Equiv.Perm.cycleType g
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ ∃ a ∈ (Equiv.Perm.cycleFactorsFinset g).val, (Finset.card ∘ Equiv.Perm.support) a = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ (Equiv.Perm.support c).card ∈ Equiv.Perm.cycleType g
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | use c | case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ ∃ a ∈ (Equiv.Perm.cycleFactorsFinset g).val, (Finset.card ∘ Equiv.Perm.support) a = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ c ∈ (Equiv.Perm.cycleFactorsFinset g).val ∧ (Finset.card ∘ Equiv.Perm.support) c = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ ∃ a ∈ (Equiv.Perm.cycleFactorsFinset g).val, (Finset.card ∘ Equiv.Perm.support) a = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact ⟨hc, rfl⟩ | case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ c ∈ (Equiv.Perm.cycleFactorsFinset g).val ∧ (Finset.card ∘ Equiv.Perm.support) c = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | 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
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ c ∈ (Equiv.Perm.cycleFactorsFinset g).val ∧ (Finset.card ∘ Equiv.Perm.support) c = (Equiv.Perm.support c).card
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [Equiv.Perm.mem_cycleFactorsFinset_iff] at hc | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc✝ : c ∈ Equiv.Perm.cycleFactorsFinset g
hc : Equiv.Perm.IsCycle c ∧ ∀ a ∈ Equiv.Perm.support c, c a = g a
a✝ : { val := c, property := hc✝ } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc✝ } ^ x) k = ↑(uv { val := c, property := hc✝ })
⊢ Equiv.Perm.IsCycle c | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc : c ∈ Equiv.Perm.cycleFactorsFinset g
a✝ : { val := c, property := hc } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc } ^ x) k = ↑(uv { val := c, property := hc })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact hc.left | case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc✝ : c ∈ Equiv.Perm.cycleFactorsFinset g
hc : Equiv.Perm.IsCycle c ∧ ∀ a ∈ Equiv.Perm.support c, c a = g a
a✝ : { val := c, property := hc✝ } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc✝ } ^ x) k = ↑(uv { val := c, property := hc✝ })
⊢ Equiv.Perm.IsCycle c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_5
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
c : Equiv.Perm α
hc✝ : c ∈ Equiv.Perm.cycleFactorsFinset g
hc : Equiv.Perm.IsCycle c ∧ ∀ a ∈ Equiv.Perm.support c, c a = g a
a✝ : { val := c, property := hc✝ } ∈ Finset.univ
k : ℤ
hk : (fun x => ↑{ val := c, property := hc✝ } ^ x) k = ↑(uv { val := c, property := hc✝ })
⊢ Equiv.Perm.IsCycle c
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | apply symm | case h.e'_3
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ 1 = Equiv.Perm.sign y | case h.e'_3.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Equiv.Perm.sign y = 1 | 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
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ 1 = Equiv.Perm.sign y
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | convert Equiv.Perm.sign_one | case h.e'_3.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Equiv.Perm.sign y = 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ y = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Equiv.Perm.sign y = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [← Equiv.Perm.card_support_le_one] | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ y = 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.support y).card ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ y = 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | apply le_trans (Finset.card_le_univ _) | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.support y).card ≤ 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ (Equiv.Perm.support y).card ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | change Fintype.card (MulAction.fixedBy α g) ≤ 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) ≤ 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(MulAction.fixedBy α g) ≤ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(Function.fixedPoints ⇑g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [OnCycleFactors.Equiv.Perm.card_fixedBy g, tsub_le_iff_left] | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(MulAction.fixedBy α g) ≤ 1 | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card ↑(MulAction.fixedBy α g) ≤ 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | exact h_fixed | case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.h.e'_6
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
y : Equiv.Perm ↑(Function.fixedPoints ⇑g)
uv : (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }) → ↥(Subgroup.zpowers ↑c)
hx : (θ g) (y, uv) ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
hx' : (θ g) (y, uv) ∈ Set.range ⇑(θ g)
⊢ Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
TACTIC:
|
https://github.com/AntoineChambert-Loir/Jordan4.git | d49910c127be01229697737a55a2d756e908d3e1 | Jordan/ConjClassCount.lean | OnCycleFactors.kerφ_le_alternating_iff | [3560, 1] | [3620, 51] | rw [← OnCycleFactors.hφ_ker_eq_θ_range, this] | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ x ∈ Set.range ⇑(θ g) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ ConjAct.toConjAct x ∈ Subgroup.map (Subgroup.subtype (MulAction.stabilizer (ConjAct (Equiv.Perm α)) g)) ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
h_odd : ∀ i ∈ Equiv.Perm.cycleType g, Odd i
h_fixed : Fintype.card α ≤ Multiset.sum (Equiv.Perm.cycleType g) + 1
h_count : ∀ (i : ℕ), Multiset.count i (Equiv.Perm.cycleType g) ≤ 1
x : ConjAct (Equiv.Perm α)
hx : x ∈ MulAction.stabilizer (ConjAct (Equiv.Perm α)) g
this : MonoidHom.ker (φ g) = ⊤
⊢ x ∈ Set.range ⇑(θ g)
TACTIC:
|
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