awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ Eq.{u_2 + 1} (α := A)
    ((ϕ : (a : ↥H) → A)
        (List.prod
          (List.map
            (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
              Subtype.mk (p := fun (x : G) ↦ x ∈ H)
                ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out *
                  Quotient.out q.out)
                ⋯)
            (Finset.toList Finset.univ))) :
      A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
have refine : Eq.{u_1 + 1} (α := G)
  ((fun (a : ↥H) ↦ (↑a : G))
    (List.prod
      (List.map
        (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
          Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out)
            sorry)
        (Finset.toList Finset.univ))))
  ((fun (a : ↥H) ↦ (↑a : G)) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) sorry))
Tactic: refine congrArg ϕ (Subtype.coe_injective ?_)
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ Eq.{u_1 + 1} (α := G)
    ((fun (a : ↥H) ↦ (↑a : G))
      (List.prod
        (List.map
          (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
            Subtype.mk (p := fun (x : G) ↦ x ∈ H)
              ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out)
              ⋯)
          (Finset.toList Finset.univ))))
    ((fun (a : ↥H) ↦ (↑a : G)) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯))
have dsimp : (↑(List.prod
        (List.map
          (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
            Subtype.mk (p := fun (x : G) ↦ x ∈ H)
              ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out)
              sorry)
          (Finset.toList Finset.univ))) :
    G) =
  g ^ H.index
Tactic: dsimp only
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ (↑(List.prod
          (List.map
            (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
              Subtype.mk (p := fun (x : G) ↦ x ∈ H)
                ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out *
                  Quotient.out q.out)
                ⋯)
            (Finset.toList Finset.univ))) :
      G) =
    g ^ H.index
have rw : Eq.{u_1 + 1} (α := G)
  (↑(List.prod
        (List.map
          (fun
              (q :
                Quotient (α := G ⧸ H)
                  (orbitRel
                    (↥(zpowers
                        (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry) : G)))
                    (G ⧸ H))) ↦
            Subtype.mk (p := fun (x : G) ↦ x ∈ H)
              ((Quotient.out q.out)⁻¹ *
                  (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry) : G) ^
                    Function.minimalPeriod
                      (fun (x : G ⧸ H) ↦
                        (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry) : G) • x)
                      q.out *
                Quotient.out q.out)
              sorry)
          (Finset.toList Finset.univ))) :
    G)
  (↑(List.prod
        (List.map
          (fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
            HPow.hPow (α := ↥(zpowers g))
              (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry)
              (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
          (Finset.toList Finset.univ))) :
    G)
Tactic: rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,
      index_eq_sum_minimalPeriod H g, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_map_toList]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ Eq.{u_1 + 1} (α := G)
    (↑(List.prod
          (List.map
            (fun
                (q :
                  Quotient (α := G ⧸ H)
                    (orbitRel
                      (↥(zpowers
                          (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯) : G)))
                      (G ⧸ H))) ↦
              Subtype.mk (p := fun (x : G) ↦ x ∈ H)
                ((Quotient.out q.out)⁻¹ *
                    (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯) : G) ^
                      Function.minimalPeriod
                        (fun (x : G ⧸ H) ↦
                          (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯) : G) •
                            x)
                        q.out *
                  Quotient.out q.out)
                ⋯)
            (Finset.toList Finset.univ))) :
      G)
    (↑(List.prod
          (List.map
            (fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
              HPow.hPow (α := ↥(zpowers g))
                (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯)
                (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
            (Finset.toList Finset.univ))) :
      G)
have simp : Eq.{u_1 + 1} (α := G)
  (List.prod
    (List.map (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
      (Function.comp (β := ↥H) (Subtype.val (p := fun (x : G) ↦ x ∈ H))
        fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
        Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out) sorry)
      (Finset.toList Finset.univ)))
  (List.prod
    (List.map (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
      (Function.comp (β := ↥(zpowers g))
        (Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x))
        fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
        HPow.hPow (α := ↥(zpowers g))
          (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry)
          (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
      (Finset.toList Finset.univ)))
Tactic: simp only [Subgroup.val_list_prod, List.map_map]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ Eq.{u_1 + 1} (α := G)
    (List.prod
      (List.map (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
        (Function.comp (β := ↥H) (Subtype.val (p := fun (x : G) ↦ x ∈ H))
          fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
          Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out)
            ⋯)
        (Finset.toList Finset.univ)))
    (List.prod
      (List.map (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
        (Function.comp (β := ↥(zpowers g))
          (Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x))
          fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
          HPow.hPow (α := ↥(zpowers g))
            (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯)
            (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
        (Finset.toList Finset.univ)))
have e_xs : Eq.{u_1 + 1} (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → G)
  (Function.comp (β := ↥H) (Subtype.val (p := fun (x : G) ↦ x ∈ H))
    fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
    Subtype.mk (p := fun (x : G) ↦ x ∈ H)
      ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out) sorry)
  (Function.comp (β := ↥(zpowers g)) (Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x))
    fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
    HPow.hPow (α := ↥(zpowers g)) (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry)
      (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
Tactic: congr 2
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
key :
  ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
    (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k
⊢ Eq.{u_1 + 1} (α := Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → G)
    (Function.comp (β := ↥H) (Subtype.val (p := fun (x : G) ↦ x ∈ H))
      fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
      Subtype.mk (p := fun (x : G) ↦ x ∈ H)
        ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out) ⋯)
    (Function.comp (β := ↥(zpowers g)) (Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x))
      fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
      HPow.hPow (α := ↥(zpowers g)) (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g ⋯)
        (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
have e_xs : ∀ (x : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
  Eq (α := G)
    (Function.comp (β := ↥H) (Subtype.val (p := fun (x : G) ↦ x ∈ H))
      (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
        Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out) sorry)
      x)
    (Function.comp (β := ↥(zpowers g)) (Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x))
      (fun (i : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
        HPow.hPow (α := ↥(zpowers g))
          (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (zpowers g) x) g sorry)
          (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) i.out))
      x)
Tactic: funext
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
letI letI₁ letI₂ :
  Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
⊢ Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
have letI₃ : ∀
  (key :
    ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
      (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
  Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) sorry) : A)
Tactic: classical
    letI := H.fintypeQuotientOfFiniteIndex
    change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key
    rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_map_toList,
      ← Function.comp_def ϕ, List.prod_map_hom]
    refine congrArg ϕ (Subtype.coe_injective ?_)
    dsimp only
    rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,
      index_eq_sum_minimalPeriod H g, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_map_toList]
    simp only [Subgroup.val_list_prod, List.map_map]
    congr 2
    funext
    apply key
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
letI letI₁ letI₂ :
  Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
letI₃ :
  ∀
    (key :
      ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
        (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
    Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
      ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
⊢ Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
have letI₄ : let this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex;
∀
  (key :
    ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
      (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
  Eq.{u_2 + 1} (α := A)
    ((ϕ : (a : ↥H) → A)
        (List.prod
          (List.map
            (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
              Subtype.mk (p := fun (x : G) ↦ x ∈ H)
                ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out *
                  Quotient.out q.out)
                sorry)
            (Finset.toList Finset.univ))) :
      A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) sorry) : A)
Tactic: classical
    letI := H.fintypeQuotientOfFiniteIndex
    change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key
    rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_map_toList,
      ← Function.comp_def ϕ, List.prod_map_hom]
    refine congrArg ϕ (Subtype.coe_injective ?_)
    dsimp only
    rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,
      index_eq_sum_minimalPeriod H g, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_map_toList]
    simp only [Subgroup.val_list_prod, List.map_map]
    congr 2
    funext
    apply key
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
inst_2 : H.FiniteIndex
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
letI letI₁ letI₂ :
  Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
letI₃ :
  ∀
    (key :
      ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
        (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
    Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
      ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
letI₄ :
  let this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex;
  ∀
    (key :
      ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
        (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
    Eq.{u_2 + 1} (α := A)
      ((ϕ : (a : ↥H) → A)
          (List.prod
            (List.map
              (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
                Subtype.mk (p := fun (x : G) ↦ x ∈ H)
                  ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out *
                    Quotient.out q.out)
                  ⋯)
              (Finset.toList Finset.univ))) :
        A)
      ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
⊢ Eq.{u_2 + 1} (α := A) ((ϕ.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
have letI₅ : let this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex;
∀
  (key :
    ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H),
      (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g₀⁻¹ * g ^ k * g₀) hk) : G) = g ^ k),
  Eq.{u_1 + 1} (α := G)
    ((fun (a : ↥H) ↦ (↑a : G))
      (List.prod
        (List.map
          (fun (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))) ↦
            Subtype.mk (p := fun (x : G) ↦ x ∈ H)
              ((Quotient.out q.out)⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q.out * Quotient.out q.out)
              sorry)
          (Finset.toList Finset.univ))))
    ((fun (a : ↥H) ↦ (↑a : G)) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) sorry))
Tactic: classical
    letI := H.fintypeQuotientOfFiniteIndex
    change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key
    rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_map_toList,
      ← Function.comp_def ϕ, List.prod_map_hom]
    refine congrArg ϕ (Subtype.coe_injective ?_)
    dsimp only
    rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,
      index_eq_sum_minimalPeriod H g, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_map_toList]
    simp only [Subgroup.val_list_prod, List.map_map]
    congr 2
    funext
    apply key
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : FiniteIndex (center G)
g : G
k : ℕ
x : G
hk : Membership.mem (γ := Subgroup G) (center G) (x⁻¹ * g ^ k * x)
⊢ x⁻¹ * g ^ k * x = g ^ k
have rw : x⁻¹ * (x⁻¹ * g ^ k * x) = x⁻¹ * g ^ k
Tactic: rw [← mul_right_inj, ← hk.comm,
    mul_inv_cancel_right]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : FiniteIndex (center G)
g : G
k : ℕ
x : G
hk : Membership.mem (γ := Subgroup G) (center G) (x⁻¹ * g ^ k * x)
rw : x⁻¹ * (x⁻¹ * g ^ k * x) = x⁻¹ * g ^ k
a : G
⊢ x⁻¹ * g ^ k * x = g ^ k
have rw₁ : x⁻¹ * g ^ k * x * x⁻¹ = x⁻¹ * g ^ k
Tactic: rw [← mul_right_inj, ← hk.comm,
    mul_inv_cancel_right]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : FiniteIndex (center G)
g : G
k : ℕ
x : G
hk : Membership.mem (γ := Subgroup G) (center G) (x⁻¹ * g ^ k * x)
rw : x⁻¹ * (x⁻¹ * g ^ k * x) = x⁻¹ * g ^ k
a : G
rw₁ : x⁻¹ * g ^ k * x * x⁻¹ = x⁻¹ * g ^ k
rw₂ : G
⊢ x⁻¹ * g ^ k * x = g ^ k
have rw₃ : x⁻¹ * g ^ k = x⁻¹ * g ^ k
Tactic: rw [← mul_right_inj, ← hk.comm,
    mul_inv_cancel_right]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
A : Type u_2
inst_1 : CommGroup A
ϕ : MonoidHom.{u_1, u_2} (↥H) A
T : H.LeftTransversal
inst_2 : FiniteIndex (center G)
a b : G
⊢ Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (center G) x) ((a * b) ^ index (center G)) ⋯ =
    Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (center G) x) (a ^ index (center G)) ⋯ *
      Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (center G) x) (b ^ index (center G)) ⋯
have simp_rw : (transfer (id ↥(center G)) : (a : G) → ↥(center G)) (a * b) =
  HMul.hMul (α := ↥(center G)) (β := ↥(center G)) ((transfer (id ↥(center G)) : (a : G) → ↥(center G)) a : ↥(center G))
    ((transfer (id ↥(center G)) : (a : G) → ↥(center G)) b : ↥(center G))
Tactic: simp_rw [← show ∀ _, (_ : center G) = _ from transfer_center_eq_pow, map_mul]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
p : ℕ
P : Sylow p G
hP : LE.le (α := Subgroup G) (normalizer (↑P : Set G)) (centralizer (↑P : Set G))
inst_1 : Fact (Nat.Prime p)
inst_2 : Finite.{u_1 + 1} (Sylow p G)
g : G
hg : g ∈ P
k : ℕ
g₀ : G
h : g₀⁻¹ * g ^ k * g₀ ∈ P
⊢ g₀⁻¹ * g ^ k * g₀ = g ^ k
have this : IsMulCommutative.{u_1} ↥(↑P : Subgroup G)
Tactic: haveI : IsMulCommutative P :=
    ⟨⟨fun a b => Subtype.ext (hP (le_normalizer b.2) a a.2)⟩⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
p : ℕ
P : Sylow p G
hP : LE.le (α := Subgroup G) (normalizer (↑P : Set G)) (centralizer (↑P : Set G))
inst_1 : Fact (Nat.Prime p)
inst_2 : Finite.{u_1 + 1} (Sylow p G)
g : G
hg : g ∈ P
k : ℕ
g₀ : G
h : g₀⁻¹ * g ^ k * g₀ ∈ P
this : IsMulCommutative.{u_1} ↥(↑P : Subgroup G)
⊢ g₀⁻¹ * g ^ k * g₀ = g ^ k
have hg₁ : g ^ k ∈ (↑P : Subgroup G)
Tactic: replace hg := P.pow_mem hg k
Cleared hypotheses: hg
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
p : ℕ
P : Sylow p G
hP : LE.le (α := Subgroup G) (normalizer (↑P : Set G)) (centralizer (↑P : Set G))
inst_1 : Fact (Nat.Prime p)
inst_2 : Finite.{u_1 + 1} (Sylow p G)
g : G
k : ℕ
g₀ : G
h : g₀⁻¹ * g ^ k * g₀ ∈ P
this : IsMulCommutative.{u_1} ↥(↑P : Subgroup G)
hg : g ^ k ∈ (↑P : Subgroup G)
⊢ g₀⁻¹ * g ^ k * g₀ = g ^ k
have obtain : ∀ (n : G) (hn : Membership.mem (γ := Subgroup G) (normalizer (↑P : Set G)) n) (h : g₀⁻¹ * g ^ k * g₀ = n⁻¹ * g ^ k * n),
  g₀⁻¹ * g ^ k * g₀ = g ^ k
Tactic: obtain ⟨n, hn, h⟩ := P.conj_eq_normalizer_conj_of_mem (g ^ k) g₀ hg h
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
p : ℕ
P : Sylow p G
hP : LE.le (α := Subgroup G) (normalizer (↑P : Set G)) (centralizer (↑P : Set G))
inst_1 : Fact (Nat.Prime p)
inst_2 : Finite.{u_1 + 1} (Sylow p G)
inst_3 : (↑P : Subgroup G).FiniteIndex
⊢ (transferSylow P hP).ker.IsComplement' (↑P : Subgroup G)
have hf : Function.Bijective (α := ↥(↑P : Subgroup G)) (β := ↥(↑P : Subgroup G))
  (⇑((transferSylow P hP).restrict (↑P : Subgroup G)) : (a : ↥(↑P : Subgroup G)) → ↥(↑P : Subgroup G))
Tactic: have hf : Function.Bijective ((transferSylow P hP).restrict (P : Subgroup G)) :=
    (transferSylow_restrict_eq_pow P hP).symm ▸ (P.2.powEquiv' P.not_dvd_index).bijective
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
p : ℕ
P : Sylow p G
hP : LE.le (α := Subgroup G) (normalizer (↑P : Set G)) (centralizer (↑P : Set G))
inst_1 : Fact (Nat.Prime p)
inst_2 : Finite.{u_1 + 1} (Sylow p G)
inst_3 : (↑P : Subgroup G).FiniteIndex
hf :
  Function.Bijective (α := ↥(↑P : Subgroup G)) (β := ↥(↑P : Subgroup G))
    (⇑((transferSylow P hP).restrict (↑P : Subgroup G)) : (a : ↥(↑P : Subgroup G)) → ↥(↑P : Subgroup G))
⊢ (transferSylow P hP).ker.IsComplement' (↑P : Subgroup G)
have hf₁ : Function.Injective (α := ↥(↑P : Subgroup G)) (β := ↥(↑P : Subgroup G))
    (⇑((transferSylow P hP).restrict (↑P : Subgroup G)) : (a : ↥(↑P : Subgroup G)) → ↥(↑P : Subgroup G)) ∧
  map (transferSylow P hP) (↑P : Subgroup G) = ⊤
Tactic: rw [Function.Bijective, ← range_eq_top, restrict_range] at hf
Cleared hypotheses: hf