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
T : H.LeftTransversal := default
g h : G
rw : diff ϕ T (g • h • T) = diff ϕ T (g • T) * diff ϕ T (h • T)
rw₁ : diff ϕ T (g • T) * diff ϕ (g • T) (g • h • T) = diff ϕ T (g • T) * diff ϕ T (h • T)
S : H.LeftTransversal
⊢ diff ϕ T ((g * h) • T) = diff ϕ T (g • T) * diff ϕ T (h • T)
have rw₂ : diff ϕ T (g • T) * diff ϕ T (h • T) = diff ϕ T (g • T) * diff ϕ T (h • T)
Tactic: rw [mul_smul, ← diff_mul_diff, smul_diff_smul]
-- 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 : H.FiniteIndex
g : G
⊢ (ϕ.transfer : (a : G) → A) g = diff ϕ T (g • T)
have rw : ({ toFun := fun (g : G) ↦ diff ϕ default (HSMul.hSMul (β := H.LeftTransversal) g default), map_one' := sorry,
        map_mul' := sorry } :
      (a : G) → A)
    g =
  diff ϕ default (HSMul.hSMul (β := H.LeftTransversal) g default)
Tactic: rw [transfer, ← diff_mul_diff, ← smul_diff_smul, mul_comm, diff_mul_diff]
-- 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
inst_3 : Fintype (Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ Eq.{u_2 + 1} (α := A)
    (∏ s : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
        (ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            (HMul.hMul (α := G)
              (↑((⋯.leftQuotientEquiv (S := (↑(H.transferTransversal g) : Set G)) :
                        (a : G ⧸ H) → (↑(↑(H.transferTransversal g) : Set G) : Type u_1))
                      (((H.quotientEquivSigmaZMod g).symm :
                          (a :
                              (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ×
                                ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) →
                            G ⧸ H)
                        (Sigma.mk (β := fun (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ↦
                          ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) q s))) :
                  G)⁻¹
              (↑((⋯.leftQuotientEquiv (S := (↑(g • H.transferTransversal g) : Set G)) :
                      (a : G ⧸ H) → (↑(↑(g • H.transferTransversal g) : Set G) : Type u_1))
                    (((H.quotientEquivSigmaZMod g).symm :
                        (a :
                            (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ×
                              ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) →
                          G ⧸ H)
                      (Sigma.mk (β := fun (q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)) ↦
                        ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) q s))) :
                G))
            ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)
have simp : Eq.{u_2 + 1} (α := A)
  (∏ x : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
      (ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))⁻¹ *
            if x = 0 then
              g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
            else HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))
          sorry) :
    A)
  ((ϕ : (a : ↥H) → A)
      (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
        ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
          Quotient.out (Quotient.out q))
        sorry) :
    A)
Tactic: simp only [quotientEquivSigmaZMod_symm_apply, transferTransversal_apply',
          transferTransversal_apply'']
-- 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
inst_3 : Fintype (Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
⊢ Eq.{u_2 + 1} (α := A)
    (∏ x : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
        (ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then
                g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))
            ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)
have rw : Eq.{u_2 + 1} (α := A)
  ((ϕ : (a : ↥H) → A)
      (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
        ((HPow.hPow (β := ℤ) g (ZMod.cast (n := Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)) 0) *
              Quotient.out (Quotient.out q))⁻¹ *
          if Eq (α := ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) 0 0 then
            g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
          else
            HPow.hPow (β := ℤ) g
                (ZMod.cast (n := Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)) 0) *
              Quotient.out (Quotient.out q))
        sorry) :
    A)
  ((ϕ : (a : ↥H) → A)
      (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
        ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
          Quotient.out (Quotient.out q))
        sorry) :
    A)
Tactic: rw [Fintype.prod_eq_single (0 : ZMod (Function.minimalPeriod (g • ·) q.out)) _]
-- 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
inst_3 : Fintype (Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
rw :
  Eq.{u_2 + 1} (α := A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((HPow.hPow (β := ℤ) g
                  (ZMod.cast (n := Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)) 0) *
                Quotient.out (Quotient.out q))⁻¹ *
            if Eq (α := ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))) 0 0 then
              g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
            else
              HPow.hPow (β := ℤ) g
                  (ZMod.cast (n := Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)) 0) *
                Quotient.out (Quotient.out q))
          ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)
⊢ Eq.{u_2 + 1} (α := A)
    (∏ x : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q)),
        (ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then
                g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))
            ⋯) :
      A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((Quotient.out (Quotient.out q))⁻¹ * g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) *
            Quotient.out (Quotient.out q))
          ⋯) :
      A)
have rw₁ : ∀ (x : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))),
  x ≠ 0 →
    Eq.{u_2 + 1} (α := A)
      ((ϕ : (a : ↥H) → A)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
            ((HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))⁻¹ *
              if x = 0 then
                g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
              else HPow.hPow (β := ℤ) g x.cast * Quotient.out (Quotient.out q))
            sorry) :
        A)
      (1 : A)
Tactic: rw [Fintype.prod_eq_single (0 : ZMod (Function.minimalPeriod (g • ·) q.out)) _]
-- 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
inst_3 : Fintype (Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)))
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
q : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)
k : ZMod (Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q))
hk : k ≠ 0
⊢ Eq.{u_2 + 1} (α := A)
    ((ϕ : (a : ↥H) → A)
        (Subtype.mk (p := fun (x : G) ↦ x ∈ H)
          ((HPow.hPow (β := ℤ) g k.cast * Quotient.out (Quotient.out q))⁻¹ *
            if k = 0 then
              g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) (Quotient.out q) * Quotient.out (Quotient.out q)
            else HPow.hPow (β := ℤ) g k.cast * Quotient.out (Quotient.out q))
          ⋯) :
      A)
    (1 : A)
have simp : Eq.{u_2 + 1} (α := A) ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) 1 sorry) : A) (1 : A)
Tactic: simp only [if_neg hk, inv_mul_cancel]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
⊢ g ^ H.index ∈ H
have pos : ∀ (hH : H.index = 0), g ^ H.index ∈ H
Tactic: by_cases hH : H.index = 0
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
pos : ∀ (hH : H.index = 0), g ^ H.index ∈ H
⊢ g ^ H.index ∈ H
have neg : ∀ (hH : ¬H.index = 0), g ^ H.index ∈ H
Tactic: by_cases hH : H.index = 0
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : H.index = 0
⊢ g ^ H.index ∈ H
have pos : 1 ∈ H
Tactic: rw [hH, pow_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
⊢ g ^ H.index ∈ H
have key₁ : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H
Tactic: replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
      (congr_arg (· ∈ H) (key k g₀ hk)).mp hk
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H
⊢ g ^ H.index ∈ H
have key₁ : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H
Tactic: replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
      key (Function.minimalPeriod (g • ·) q) q.out
        (QuotientGroup.out_conj_pow_minimalPeriod_mem H g q)
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H
f : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
  fun (q : 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) q.out)
⊢ g ^ H.index ∈ H
have hf : ∀ (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
  Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g)) (f q)
Tactic: have hf : ∀ q, f q ∈ H.subgroupOf (zpowers g) := fun q => key q.out
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H
f : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
  fun (q : 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) q.out)
hf :
  ∀ (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
    Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g)) (f q)
⊢ g ^ H.index ∈ H
have key₁ : Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g))
  (∏ c : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)), f c)
Tactic: replace key :=
      Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
⊢ g ^ H.index ∈ H
have neg : ∀ (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H), g ^ H.index ∈ H
Tactic: classical
    replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
      (congr_arg (· ∈ H) (key k g₀ hk)).mp hk
    replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
      key (Function.minimalPeriod (g • ·) q) q.out
        (QuotientGroup.out_conj_pow_minimalPeriod_mem H g q)
    let f : Quotient (orbitRel (zpowers g) (G ⧸ H)) → zpowers g := fun q =>
      (⟨g, mem_zpowers g⟩ : zpowers g) ^ Function.minimalPeriod (g • ·) q.out
    have hf : ∀ q, f q ∈ H.subgroupOf (zpowers g) := fun q => key q.out
    replace key :=
      Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q
    simpa only [f, Finset.prod_pow_eq_pow_sum, index_eq_sum_minimalPeriod H g] using key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
neg neg₁ neg₂ : ∀ (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H), g ^ H.index ∈ H
⊢ g ^ H.index ∈ H
have neg₃ : ∀ (key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H), g ^ H.index ∈ H
Tactic: classical
    replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
      (congr_arg (· ∈ H) (key k g₀ hk)).mp hk
    replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
      key (Function.minimalPeriod (g • ·) q) q.out
        (QuotientGroup.out_conj_pow_minimalPeriod_mem H g q)
    let f : Quotient (orbitRel (zpowers g) (G ⧸ H)) → zpowers g := fun q =>
      (⟨g, mem_zpowers g⟩ : zpowers g) ^ Function.minimalPeriod (g • ·) q.out
    have hf : ∀ q, f q ∈ H.subgroupOf (zpowers g) := fun q => key q.out
    replace key :=
      Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q
    simpa only [f, Finset.prod_pow_eq_pow_sum, index_eq_sum_minimalPeriod H g] using key
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
neg neg₁ neg₂ : ∀ (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H), g ^ H.index ∈ H
neg₃ neg₄ : ∀ (key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H), g ^ H.index ∈ H
⊢ g ^ H.index ∈ H
have neg₅ : ∀ (key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H),
  let f : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
    fun (q : 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) q.out);
  ∀
    (hf :
      ∀ (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
        Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g)) (f q)),
    g ^ H.index ∈ H
Tactic: classical
    replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
      (congr_arg (· ∈ H) (key k g₀ hk)).mp hk
    replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
      key (Function.minimalPeriod (g • ·) q) q.out
        (QuotientGroup.out_conj_pow_minimalPeriod_mem H g q)
    let f : Quotient (orbitRel (zpowers g) (G ⧸ H)) → zpowers g := fun q =>
      (⟨g, mem_zpowers g⟩ : zpowers g) ^ Function.minimalPeriod (g • ·) q.out
    have hf : ∀ q, f q ∈ H.subgroupOf (zpowers g) := fun q => key q.out
    replace key :=
      Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q
    simpa only [f, Finset.prod_pow_eq_pow_sum, index_eq_sum_minimalPeriod H g] using key
Cleared hypotheses: key
-- Lean 4 proof step from Mathlib.GroupTheory.Transfer (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
g : G
key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k
hH : ¬H.index = 0
this : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH
neg neg₁ neg₂ : ∀ (key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H), g ^ H.index ∈ H
neg₃ neg₄ : ∀ (key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H), g ^ H.index ∈ H
neg₅ :
  ∀ (key : ∀ (q : G ⧸ H), g ^ Function.minimalPeriod (fun (x : G ⧸ H) ↦ g • x) q ∈ H),
    let f : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
      fun (q : 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) q.out);
    ∀
      (hf :
        ∀ (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
          Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g)) (f q)),
      g ^ H.index ∈ H
⊢ g ^ H.index ∈ H
have neg₆ : let f : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H)) → ↥(zpowers g) :=
  fun (q : 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) q.out);
∀
  (hf :
    ∀ (q : Quotient (α := G ⧸ H) (orbitRel (↥(zpowers g)) (G ⧸ H))),
      Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g)) (f q))
  (key :
    Membership.mem (γ := Subgroup ↥(zpowers g)) (H.subgroupOf (zpowers g))
      (∏ c : Quotient (orbitRel (↥(zpowers g)) (G ⧸ H)), f c)),
  g ^ H.index ∈ H
Tactic: classical
    replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
      (congr_arg (· ∈ H) (key k g₀ hk)).mp hk
    replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
      key (Function.minimalPeriod (g • ·) q) q.out
        (QuotientGroup.out_conj_pow_minimalPeriod_mem H g q)
    let f : Quotient (orbitRel (zpowers g) (G ⧸ H)) → zpowers g := fun q =>
      (⟨g, mem_zpowers g⟩ : zpowers g) ^ Function.minimalPeriod (g • ·) q.out
    have hf : ∀ q, f q ∈ H.subgroupOf (zpowers g) := fun q => key q.out
    replace key :=
      Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q ∈ Finset.univ) => hf q
    simpa only [f, Finset.prod_pow_eq_pow_sum, index_eq_sum_minimalPeriod H g] using 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
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
⊢ 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 change : ∀
  (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: change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at 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
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.transfer : (a : G) → A) g : A)
    ((ϕ : (a : ↥H) → A) (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g ^ H.index) ⋯) : A)
have rw : 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: rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_map_toList,
      ← Function.comp_def ϕ, List.prod_map_hom]