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

Goal state before:
pp: case refine_2
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
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : N ≤ K
hH' : comap (QuotientGroup.mk' K) H = comap (QuotientGroup.mk' K) ⊤
⊢ False

Tactic:
]

Goal state after:
case refine_2
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
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : N ≤ K
hH' : comap (QuotientGroup.mk' K) H = comap (QuotientGroup.mk' K) ⊤
⊢ False
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⟩
hk_cT : (↑k).cycleType = Multiset.replicate (↑k).cycleType.card 2
this : (↑k).cycleType.card * 2 = #(↑k).support
H : ⟨c, hc⟩ ≠ ⟨d, hd⟩
⊢ #(↑⟨c, hc⟩).support + #(↑⟨d, hd⟩).support = #c.support * 2

Tactic:
hm,

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
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
hk_cT : (↑k).cycleType = Multiset.replicate (↑k).cycleType.card 2
this : (↑k).cycleType.card * 2 = #(↑k).support
H : ⟨c, hc⟩ ≠ ⟨d, hd⟩
⊢ #d.support + #(↑⟨d, hd⟩).support = #d.support * 2
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: case pos.H2
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
m : Multiset ℕ
hm : m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a
⊢ (Fintype.card α)! = (Fintype.card α)!

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⟩
hk_cT : (↑k).cycleType = Multiset.replicate (↑k).cycleType.card 2
this : (↑k).cycleType.card * 2 = #(↑k).support
H : ⟨c, hc⟩ ≠ ⟨d, hd⟩
⊢ #d.support + #(↑⟨d, hd⟩).support = #d.support + #d.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 ↥N
inst✝ : (map N.subtype K).Normal
key : map N.subtype K = map N.subtype ⊥ ∨ map N.subtype K = N
⊢ K = ⊥ ∨ 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 ↥N
inst✝ : (map N.subtype K).Normal
key : map N.subtype K = map N.subtype ⊥ ∨ map N.subtype K = N
⊢ K = ⊥ ∨ K = ⊤
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⟩
hk_cT : (↑k).cycleType = Multiset.replicate (↑k).cycleType.card 2
this : (↑k).cycleType.card * 2 = #(↑k).support
that : (↑k).cycleType.card = #c.support
⊢ #c.support ∈ g.cycleType

Tactic:
]

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
hτ : τ ∈ range_toPermHom' g
k : ↥(Subgroup.centralizer {g}) := a.toCentralizer ⟨τ, hτ⟩
hk : k = a.toCentralizer ⟨τ, hτ⟩
hk_cT : (↑k).cycleType = Multiset.replicate (↑k).cycleType.card 2
this : (↑k).cycleType.card * 2 = #(↑k).support
that : (↑k).cycleType.card = #c.support
⊢ #c.support ∈ g.cycleType
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
hn' : 2 ≤ n
hα : n ≤ card α
hn₀ : n ≠ 0
⊢ n ! = n !

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 ↥N
inst✝ : (map N.subtype K).Normal
key : K = ⊥ ∨ K = ⊤
⊢ K = ⊥ ∨ 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 ↥N
inst✝ : (map N.subtype K).Normal
key : K = ⊥ ∨ K = ⊤
⊢ K = ⊥ ∨ K = ⊤
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'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝ : Finite G
this : Fact (Nat.Prime (Nat.card ↥N).minFac)
P : Sylow (Nat.card ↥N).minFac ↥N
⊢ ↑P = ⊤

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'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝ : Finite G
this : Fact (Nat.Prime (Nat.card ↥N).minFac)
P : Sylow (Nat.card ↥N).minFac ↥N
⊢ ↑P = ⊤
Module: Mathlib.GroupTheory.Perm.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
hn' : 2 ≤ n
hα : n ≤ card α
hn₀ : n ≠ 0
aux : n ! = (n - 1)! * n
⊢ #{g | g.cycleType = {n}} * (n * (card α - n)!) = (card α)!

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
n : ℕ
hn' : 2 ≤ n
hα : n ≤ card α
hn₀ : n ≠ 0
aux : n ! = (n - 1)! * n
⊢ #{g | g.cycleType = {n}} * (n * (card α - n)!) = (card α)!
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: 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)
i : ι
w₁ : Word G
h₁ : H
hw₁ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₁.toList → g ∈ d.set i
w₂ : Word G
h₂ : H
hw₂ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₂.toList → g ∈ d.set i
h : w₁ = ((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂
⊢ { toWord := w₁, head := h₁, normalized := hw₁ } = { toWord := w₂, head := h₂, normalized := hw₂ }

Tactic:
]

Goal state after:
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)
i : ι
w₁ : Word G
h₁ : H
hw₁ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₁.toList → g ∈ d.set i
w₂ : Word G
h₂ : H
hw₂ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₂.toList → g ∈ d.set i
h : w₁ = ((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂
⊢ { toWord := w₁, head := h₁, normalized := hw₁ } = { toWord := w₂, head := h₂, normalized := hw₂ }
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ g.support.card ≤ Fintype.card α

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
⊢ g.support.card ≤ Fintype.card α
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: case «3»
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : ∀ b ∈ g.cycleType, b ∣ 2 ^ n
h1 : g.cycleType.sum ≤ 4
k : ℕ
hk : 3 ∈ g.cycleType
hk4 : 3 ≤ 4
hk1 : 1 < 3
⊢ 3 = 2

Tactic:
]

Goal state after:
case «3»
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : ∀ b ∈ g.cycleType, b ∣ 2 ^ n
h1 : g.cycleType.sum ≤ 4
k : ℕ
hk : 3 ∈ g.cycleType
hk4 : 3 ≤ 4
hk1 : 1 < 3
⊢ 3 = 2
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: 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)
i : ι
h₁ : H
w₂ : Word G
h₂ : H
hw₂ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₂.toList → g ∈ d.set i
hw₁✝ :
  ∀ (i_1 : ι) (g : G i_1), ⟨i_1, g⟩ ∈ (((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂).toList → g ∈ d.set i_1
hw₁ : ∀ (i_1 : ι) (g : G i_1), ⟨i_1, g⟩ ∈ (CoprodI.of ((φ i) (h₁⁻¹ * h₂)) • w₂).toList → g ∈ d.set i_1
this : h₁ = h₂
⊢ { toWord := ((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂, head := h₁, normalized := hw₁✝ } =
    { toWord := w₂, head := h₂, normalized := hw₂ }

Tactic:
]

Goal state after:
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)
i : ι
h₁ : H
w₂ : Word G
h₂ : H
hw₂ : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w₂.toList → g ∈ d.set i
hw₁✝ :
  ∀ (i_1 : ι) (g : G i_1), ⟨i_1, g⟩ ∈ (((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂).toList → g ∈ d.set i_1
hw₁ : ∀ (i_1 : ι) (g : G i_1), ⟨i_1, g⟩ ∈ (CoprodI.of ((φ i) (h₁⁻¹ * h₂)) • w₂).toList → g ∈ d.set i_1
this : h₁ = h₂
⊢ { toWord := ((CoprodI.of ((φ i) h₁))⁻¹ * CoprodI.of ((φ i) h₂)) • w₂, head := h₁, normalized := hw₁✝ } =
    { toWord := w₂, head := h₂, normalized := hw₂ }
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : N = ⊤
hN1 : Nat.card ↥N = 0
⊢ ∃ H, N.IsComplement' H

Tactic:
]

Goal state after:
case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : N = ⊤
hN1 : Nat.card ↥N = 0
⊢ ∃ H, N.IsComplement' H
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : N = ⊤
hN1 : Nat.card ↥N = 0
⊢ ∃ H, ⊤.IsComplement' H

Tactic:
]

Goal state after:
case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : N = ⊤
hN1 : Nat.card ↥N = 0
⊢ ∃ H, ⊤.IsComplement' H
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: case «4»
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
n : ℕ
hg : orderOf g ∣ 2 ^ n
k : ℕ
hk : 4 ∈ g.cycleType
hk4 : 4 ≤ 4
hk1 : 1 < 4
hg0 : 4 ≠ 2
t : Multiset ℕ
h1 : t = Multiset.replicate t.card 0
ht : g.cycleType = 4 ::ₘ t
⊢ (-1) ^ (g.cycleType.sum + g.cycleType.card) ≠ 1

Tactic:
]

Goal state after:
case «4»
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
n : ℕ
hg : orderOf g ∣ 2 ^ n
k : ℕ
hk : 4 ∈ g.cycleType
hk4 : 4 ≤ 4
hk1 : 1 < 4
hg0 : 4 ≠ 2
t : Multiset ℕ
h1 : t = Multiset.replicate t.card 0
ht : g.cycleType = 4 ::ₘ t
⊢ (-1) ^ (g.cycleType.sum + g.cycleType.card) ≠ 1
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_count :
  ∀ x ∈ g.cycleFactorsFinset.val, ∀ y ∈ g.cycleFactorsFinset.val, (card ∘ support) x = (card ∘ support) y → x = y
x : Perm α
hx : x ∈ Subgroup.centralizer {g}
c : ↥g.cycleFactorsFinset
⊢ ↑(((toPermHom g) ⟨x, ⋯⟩) c) = ↑(1 c)

Tactic:
]

Goal state after:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_count :
  ∀ x ∈ g.cycleFactorsFinset.val, ∀ y ∈ g.cycleFactorsFinset.val, (card ∘ support) x = (card ∘ support) y → x = y
x : Perm α
hx : x ∈ Subgroup.centralizer {g}
c : ↥g.cycleFactorsFinset
⊢ ↑(((toPermHom g) ⟨x, ⋯⟩) c) = ↑(1 c)
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : Subsingleton ↥N ∧ Nonempty ↥N
hN1 : ¬Nat.card ↥N = 0
hN2 : N.index = 0
⊢ ∃ H, N.IsComplement' H

Tactic:
]

Goal state after:
case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : Subsingleton ↥N ∧ Nonempty ↥N
hN1 : ¬Nat.card ↥N = 0
hN2 : N.index = 0
⊢ ∃ H, N.IsComplement' H
Module: Mathlib.GroupTheory.PushoutI

Goal state before:
pp: ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝⁴ : Monoid K
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)
i✝ : ι
p : Pair d i✝
n : ↑↑(φ i✝).range × ↑(d.set i✝) := ⋯.equiv p.head
w : Word G :=
  (Word.equivPair i✝).symm
    (let __src := p.toPair;
    { head := ↑n.2, tail := __src.tail, fstIdx_ne := ⋯ })
i : ι
g : G i
hg :
  ⟨i, g⟩ ∈ { head := ↑n.2, tail := p.tail, fstIdx_ne := ⋯ }.tail.toList ∨
    g ≠ 1 ∧ ∃ (h : i✝ = i), g = h ▸ { head := ↑n.2, tail := p.tail, fstIdx_ne := ⋯ }.head
⊢ g ∈ d.set i

Tactic:
]

Goal state after:
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝⁴ : Monoid K
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)
i✝ : ι
p : Pair d i✝
n : ↑↑(φ i✝).range × ↑(d.set i✝) := ⋯.equiv p.head
w : Word G :=
  (Word.equivPair i✝).symm
    (let __src := p.toPair;
    { head := ↑n.2, tail := __src.tail, fstIdx_ne := ⋯ })
i : ι
g : G i
hg :
  ⟨i, g⟩ ∈ { head := ↑n.2, tail := p.tail, fstIdx_ne := ⋯ }.tail.toList ∨
    g ≠ 1 ∧ ∃ (h : i✝ = i), g = h ▸ { head := ↑n.2, tail := p.tail, fstIdx_ne := ⋯ }.head
⊢ g ∈ d.set i
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : Subsingleton ↥N ∧ Nonempty ↥N
hN1 : ¬Nat.card ↥N = 0
hN2 : N.index = 0
this : Subsingleton ↥N
⊢ ∃ H, ⊥.IsComplement' H

Tactic:
]

Goal state after:
case pos
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : Subsingleton ↥N ∧ Nonempty ↥N
hN1 : ¬Nat.card ↥N = 0
hN2 : N.index = 0
this : Subsingleton ↥N
⊢ ∃ H, ⊥.IsComplement' H
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
⊢ (∀ ⦃x : Perm α⦄, x ∈ Subgroup.centralizer {g} → x ∈ alternatingGroup α) ↔
    (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
⊢ (∀ ⦃x : Perm α⦄, x ∈ Subgroup.centralizer {g} → x ∈ alternatingGroup α) ↔
    (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
h1 : g.cycleType.sum ≤ 4
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
h1 : g.cycleType.sum ≤ 4
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}
Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : (Nat.card ↥N).Coprime N.index
hN1 : ¬Nat.card ↥N = 0
hN2 : ¬N.index = 0
⊢ Nat.card ↥N * N.index ≠ 0

Tactic:
]

Goal state after:
G : Type u
inst✝¹ : Group G
N : Subgroup G
inst✝ : N.Normal
hN : (Nat.card ↥N).Coprime N.index
hN1 : ¬Nat.card ↥N = 0
hN2 : ¬N.index = 0
⊢ Nat.card ↥N * N.index ≠ 0
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
h1 : g.cycleType.card ≤ 4 / 2
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
g : Perm α
hg0 : (-1) ^ (g.cycleType.sum + g.cycleType.card) = 1
n : ℕ
hg : orderOf g ∣ 2 ^ n
h1 : g.cycleType.card ≤ 4 / 2
h2 : g.cycleType = Multiset.replicate g.cycleType.card 2
⊢ g.cycleType = ∅ ∨ g.cycleType = {2, 2}
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_odd : ∀ c ∈ g.cycleType, Odd c
h_fixed : Fintype.card α ≤ g.cycleType.sum + 1
h_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
x : Perm α
hx : x ∈ (kerParam g).range
⊢ x ∈ alternatingGroup α

Tactic:
]

Goal state after:
case mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_odd : ∀ c ∈ g.cycleType, Odd c
h_fixed : Fintype.card α ≤ g.cycleType.sum + 1
h_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
x : Perm α
hx : x ∈ (kerParam g).range
⊢ x ∈ alternatingGroup α
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
⊢ 1 < 4

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
⊢ 1 < 4
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 / 2 = 12

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
this : Nontrivial α
⊢ Nat.factorial 4 / 2 = 12
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer

Goal state before:
pp: case mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_odd : ∀ c ∈ g.cycleType, Odd c
h_fixed : Fintype.card α ≤ g.cycleType.sum + 1
h_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
y : Perm ↑(Function.fixedPoints ⇑g)
uv : (c : ↥g.cycleFactorsFinset) → ↥(Subgroup.zpowers ↑c)
hx : (kerParam g) (y, uv) ∈ (kerParam g).range
⊢ sign y * ∏ c, sign ↑(uv c) = 1

Tactic:
]

Goal state after:
case mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
h_odd : ∀ c ∈ g.cycleType, Odd c
h_fixed : Fintype.card α ≤ g.cycleType.sum + 1
h_count : ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1
y : Perm ↑(Function.fixedPoints ⇑g)
uv : (c : ↥g.cycleFactorsFinset) → ↥(Subgroup.zpowers ↑c)
hx : (kerParam g) (y, uv) ∈ (kerParam g).range
⊢ sign y * ∏ c, sign ↑(uv c) = 1
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ 2 ^ (Nat.factorization 12) 2 = 4

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
⊢ 2 ^ (Nat.factorization 12) 2 = 4
Module: Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
this : 12 = 2 ^ 2 * 3
⊢ 2 ^ (2 * 1 + 0) = 4

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hα4 : Nat.card α = 4
S : Sylow 2 ↥(alternatingGroup α)
this : 12 = 2 ^ 2 * 3
⊢ 2 ^ (2 * 1 + 0) = 4