mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: case refine_2
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
w : Word G
i : ι
h : H
hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (CoprodI.of ((φ i) h) • w).toList → ↑(⋯.equiv g).2 = g
hhead : ↑(⋯.equiv ((Word.equivPair i) w).head).2 = ((Word.equivPair i) w).head
hh1 : ¬h = 1
this : ((Word.equivPair i) w).head = (φ i) h * ((Word.equivPair i) w).head
⊢ h = 1

Tactic:
]

Goal state after:
case refine_2
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
w : Word G
i : ι
h : H
hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (CoprodI.of ((φ i) h) • w).toList → ↑(⋯.equiv g).2 = g
hhead : ↑(⋯.equiv ((Word.equivPair i) w).head).2 = ((Word.equivPair i) w).head
hh1 : ¬h = 1
this : ((Word.equivPair i) w).head = (φ i) h * ((Word.equivPair i) w).head
⊢ h = 1
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ ↑(v ⟨g.cycleOf x, hx✝⟩) x = ↑p x

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ ↑(v ⟨g.cycleOf x, hx✝⟩) x = ↑p x
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
hn : 2 ≤ n
hα : n ≤ Fintype.card α
⊢ (if Even ({n}.sum + {n}.card) then #{g | g.cycleType = {n}} else #∅) =
    if Odd n then (n - 1)! * (Fintype.card α).choose n else 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
n : ℕ
hn : 2 ≤ n
hα : n ≤ Fintype.card α
⊢ (if Even ({n}.sum + {n}.card) then #{g | g.cycleType = {n}} else #∅) =
    if Odd n then (n - 1)! * (Fintype.card α).choose n else 0
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
h5 : Nat.card ↥K < Nat.card G
h6 : (Nat.card ↥(comap K.subtype N)).Coprime (comap K.subtype N).index
H : Subgroup ↥K
hH : (comap K.subtype N).IsComplement' H
⊢ N.index = N.index

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ ↑⟨↑p ↑⟨x, ?pos.ha✝⟩, ⋯⟩ = ↑p x

pp: case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ x ∈ (↑⟨g.cycleOf x, hx✝⟩).support

Tactic:
]

Goal state after:
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ x ∈ (↑⟨g.cycleOf x, hx✝⟩).support
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ ¬g x = x

Tactic:
]

Goal state after:
case pos.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx✝ : g.cycleOf x ∈ g.cycleFactorsFinset
hx : g x ≠ x
⊢ ¬g x = x
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
h5 : Nat.card ↥K < Nat.card G
h6 : (Nat.card ↥(comap K.subtype N)).Coprime (comap K.subtype N).index
H : Subgroup ↥K
hH : Nat.card ↥(map K.subtype H) = N.index
⊢ N.index * Nat.card ↥N = N.index * Nat.card ↥N

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ 1 + 1 + g.cycleType.sum ≤ Fintype.card α

pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ g.cycleType.sum ≤ Fintype.card α

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ 1 + 1 + g.cycleType.sum ≤ Fintype.card α

α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ g.cycleType.sum ≤ Fintype.card α
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ #g.support ≤ Fintype.card α

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
hm : 2 + g.cycleType.sum ≤ Fintype.card α
⊢ #g.support ≤ Fintype.card α
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx : g x = x
⊢ (ofSubtype u) x = ↑p x

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx : g x = x
⊢ (ofSubtype u) x = ↑p x
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
h5 : Nat.card ↥K < Nat.card G
h6 : (Nat.card ↥(comap K.subtype N)).Coprime (comap K.subtype N).index
H : Subgroup ↥K
hH : Nat.card ↥(map K.subtype H) = N.index
h7 : Nat.card ↥N * Nat.card ↥(map K.subtype H) = Nat.card G
⊢ (Nat.card ↥N).Coprime N.index

Tactic:
]

Goal state after:
G : Type u
inst✝² : Group G
N : Subgroup G
inst✝¹ : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
inst✝ : Finite G
K : Subgroup G
hK : K ⊔ N = ⊤
h3 : K ≠ ⊤
h4 : (comap K.subtype N).index = N.index
h5 : Nat.card ↥K < Nat.card G
h6 : (Nat.card ↥(comap K.subtype N)).Coprime (comap K.subtype N).index
H : Subgroup ↥K
hH : Nat.card ↥(map K.subtype H) = N.index
h7 : Nat.card ↥N * Nat.card ↥(map K.subtype H) = Nat.card G
⊢ (Nat.card ↥N).Coprime N.index
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
⊢ (Multiset.map (card ∘ support) g.cycleFactorsFinset.val).Nodup

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
⊢ (Multiset.map (card ∘ support) g.cycleFactorsFinset.val).Nodup
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
⊢ ∀ x ∈ g.cycleFactorsFinset.val, ∀ y ∈ g.cycleFactorsFinset.val, (card ∘ support) x = (card ∘ support) y → x = y

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
⊢ ∀ x ∈ g.cycleFactorsFinset.val, ∀ y ∈ g.cycleFactorsFinset.val, (card ∘ support) x = (card ∘ support) y → x = y
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
⊢ False

Tactic:
obtain ⟨a⟩ := Equiv.Perm.Basis.nonempty g

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
⊢ False
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx : g x = x
⊢ ↑⟨↑p ↑⟨x, ?neg.ha✝⟩, ⋯⟩ = ↑p x

pp: case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g

Tactic:
]

Goal state after:
case neg.ha
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
  ∀ c ∈ g.cycleFactorsFinset,
    ∃ (hc : ∀ (x : α), ↑p x ∈ c.support ↔ x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c) := fun c ↦ ⟨ofSubtype ((↑p).subtypePerm ⋯), ⋯⟩
x : α
hx : g x = x
⊢ x ∈ Function.fixedPoints ⇑g
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
x : ↥g.cycleFactorsFinset
hx : x = ⟨c, hc⟩
⊢ #(↑⟨d, hd⟩).support = #(↑⟨c, hc⟩).support

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
x : ↥g.cycleFactorsFinset
hx : x = ⟨c, hc⟩
⊢ #(↑⟨d, hd⟩).support = #(↑⟨c, hc⟩).support
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ map (centralizer {g}).subtype (toPermHom g).ker ≤ centralizer {g}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ map (centralizer {g}).subtype (toPermHom g).ker ≤ centralizer {g}
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ Fintype.card (Perm ↑(Function.fixedPoints ⇑g) × ((c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c))) =
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ Fintype.card (Perm ↑(Function.fixedPoints ⇑g) × ((c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c))) =
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ (Fintype.card α - g.cycleType.sum)! * ∏ i, Fintype.card ↥(zpowers ↑i) =
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ (Fintype.card α - g.cycleType.sum)! * ∏ i, Fintype.card ↥(zpowers ↑i) =
    (Fintype.card α - g.cycleType.sum)! * g.cycleType.prod
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: case refine_1.left
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
w : Word G
i : ι
h : H
hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (CoprodI.of ((φ i) h) • w).toList → ↑(⋯.equiv g).2 = g
hhead : ↑(⋯.equiv ((Word.equivPair i) w).head).2 = ((Word.equivPair i) w).head
hh1 : ¬h = 1
h₁ : h = 1
h₂ : ((Word.equivPair i) w).head = 1
⊢ False

Tactic:
]

Goal state after:
case refine_1.left
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
w : Word G
i : ι
h : H
hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (CoprodI.of ((φ i) h) • w).toList → ↑(⋯.equiv g).2 = g
hhead : ↑(⋯.equiv ((Word.equivPair i) w).head).2 = ((Word.equivPair i) w).head
hh1 : ¬h = 1
h₁ : h = 1
h₂ : ((Word.equivPair i) w).head = 1
⊢ False
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
x : ↥g.cycleFactorsFinset
hx : ¬x = ⟨c, hc⟩
hx' : x = ⟨d, hd⟩
⊢ #(↑⟨c, hc⟩).support = #(↑⟨d, hd⟩).support

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
x : ↥g.cycleFactorsFinset
hx : ¬x = ⟨c, hc⟩
hx' : x = ⟨d, hd⟩
⊢ #(↑⟨c, hc⟩).support = #(↑⟨d, hd⟩).support
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ ∏ x ∈ g.cycleFactorsFinset, Fintype.card ↥(zpowers x) = ∏ a ∈ g.cycleFactorsFinset, (Finset.card ∘ support) a

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
⊢ ∏ x ∈ g.cycleFactorsFinset, Fintype.card ↥(zpowers x) = ∏ a ∈ g.cycleFactorsFinset, (Finset.card ∘ support) a
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g x : Perm α
hx : x ∈ g.cycleFactorsFinset
⊢ #x.support = #x.support

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
⊢ K.index < K.index * Nat.card ↥K

Tactic:
]

Goal state after:
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
⊢ K.index < K.index * Nat.card ↥K
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case neg
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
x : ↥g.cycleFactorsFinset
hx : ¬x = ⟨c, hc⟩
hx' : ¬x = ⟨d, hd⟩
⊢ #(↑x).support = #(↑x).support

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h : Subgroup.centralizer {g} ≤ alternatingGroup α
c : Perm α
hc : c ∈ g.cycleFactorsFinset
d : Perm α
hd : d ∈ g.cycleFactorsFinset
hm : #c.support = #d.support
hm' : c ≠ d
τ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩
a : g.Basis
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
hsign_k : sign ↑k = -1
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
⊢ K ≤ N

Tactic:
]

Goal state after:
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
⊢ K ≤ N
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
k : Perm ↑(Function.fixedPoints ⇑g)
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)
⊢ ∏ i ∈ g.cycleFactorsFinset.attach, (sign.comp (zpowers ↑i).subtype) (((k, v).1, (k, v).2).2 i) =
    ∏ c ∈ g.cycleFactorsFinset.attach, sign ↑(v c)

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
k : Perm ↑(Function.fixedPoints ⇑g)
v : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)
⊢ ∏ i ∈ g.cycleFactorsFinset.attach, (sign.comp (zpowers ↑i).subtype) (((k, v).1, (k, v).2).2 i) =
    ∏ c ∈ g.cycleFactorsFinset.attach, sign ↑(v c)