awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H : Set G) R
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure S) ⊤
hU : closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ (↑(closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S))) : Set G)
r : G
hr : r ∈ R
hh : (fun (x1 x2 : G) ↦ x1 * x2) g r ∈ H
⊢ Subtype.mk (p := fun (x : G) ↦ x ∈ R) r hr = Subtype.mk (p := fun (x : G) ↦ x ∈ R) 1 hR1
have py₁ : r * (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ R) r hr) : G)⁻¹ ∈ (↑H : Set G)
Tactic: apply (isComplement_iff_existsUnique_mul_inv_mem.mp hR r).unique
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H : Set G) R
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure S) ⊤
hU : closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ (↑(closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S))) : Set G)
r : G
hr : r ∈ R
hh : (fun (x1 x2 : G) ↦ x1 * x2) g r ∈ H
py₁ : r * (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ R) r hr) : G)⁻¹ ∈ (↑H : Set G)
⊢ Subtype.mk (p := fun (x : G) ↦ x ∈ R) r hr = Subtype.mk (p := fun (x : G) ↦ x ∈ R) 1 hR1
have py₂ : r * (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ R) 1 hR1) : G)⁻¹ ∈ (↑H : Set G)
Tactic: apply (isComplement_iff_existsUnique_mul_inv_mem.mp hR r).unique
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H : Set G) R
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure S) ⊤
hU : closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ (↑(closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S))) : Set G)
r : G
hr : r ∈ R
hh : (fun (x1 x2 : G) ↦ x1 * x2) g r ∈ H
⊢ r * (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ R) r hr) : G)⁻¹ ∈ (↑H : Set G)
have py₁ : 1 ∈ (↑H : Set G)
Tactic: rw [Subtype.coe_mk, mul_inv_cancel]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H : Set G) R
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure S) ⊤
hU : closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S)) ≤ H
g : G
hg : g ∈ (↑(closure ((fun (g : G) ↦ g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) '' (R * S))) : Set G)
r : G
hr : r ∈ R
hh : (fun (x1 x2 : G) ↦ x1 * x2) g r ∈ H
⊢ r * (↑(Subtype.mk (p := fun (x : G) ↦ x ∈ R) 1 hR1) : G)⁻¹ ∈ (↑H : Set G)
have py₂ : r ∈ (↑H : Set G)
Tactic: rw [Subtype.coe_mk, inv_one, mul_one]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hR : IsComplement (↑H : Set G) R
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure S) ⊤
⊢ Eq (α := Subgroup ↥H)
    (closure
      ((fun (g : G) ↦ Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g * (↑(hR.toRightFun (T := R) g) : G)⁻¹) ⋯) '' (R * S)))
    ⊤
have rw : map H.subtype ⊤ ≤
  closure
    ((fun (x : G) ↦
        (H.subtype : (a : ↥H) → G)
          (Subtype.mk (p := fun (x : G) ↦ x ∈ H) (x * (↑(hR.toRightFun (T := R) x) : G)⁻¹) sorry)) ''
      (R * S))
Tactic: rw [eq_top_iff, ← map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : DecidableEq G
R S : Finset G
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
⊢ Eq (α := Subgroup ↥H)
    (closure
      (↑(Finset.image
            (fun (g : G) ↦ Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g * (↑(hR.toRightFun (T := (↑R : Set G)) g) : G)⁻¹) ⋯)
            (R * S)) :
        Set ↥H))
    ⊤
have rw : Eq (α := Subgroup ↥H)
  (closure
    ((fun (g : G) ↦ Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g * (↑(hR.toRightFun (T := (↑R : Set G)) g) : G)⁻¹) sorry) ''
      ((↑R : Set G) * (↑S : Set G))))
  ⊤
Tactic: rw [Finset.coe_image, Finset.coe_mul]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have haveI : ∀ (this : DecidableEq G), ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
Tactic: haveI : DecidableEq G := Classical.decEq G
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this : DecidableEq G
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have obtain : ∀ (R₀ : Set G) (hR : IsComplement (↑H : Set G) R₀) (hR1 : 1 ∈ R₀),
  ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
Tactic: obtain ⟨R₀, hR, hR1⟩ := H.exists_isComplement_right 1
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this : DecidableEq G
R₀ : Set G
hR : IsComplement (↑H : Set G) R₀
hR1 : 1 ∈ R₀
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have haveI : ∀ (this : Fintype (↑R₀ : Type u_1)),
  ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
Tactic: haveI : Fintype R₀ := Fintype.ofEquiv _ hR.rightQuotientEquiv
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
hR : IsComplement (↑H : Set G) R₀
hR1 : 1 ∈ R₀
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have hR₁ : IsComplement (↑H : Set G) (↑R : Set G)
Tactic: replace hR : IsComplement (H : Set G) R := by rwa [Set.coe_toFinset]
Cleared hypotheses: hR
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
hR1 : 1 ∈ R₀
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have hR1₁ : 1 ∈ R
Tactic: replace hR1 : (1 : G) ∈ R := by rwa [Set.mem_toFinset]
Cleared hypotheses: hR1
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
⊢ ∃ (T : Finset ↥H), #T ≤ H.index * #S ∧ Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤
have refine : Finset.card (α := ↥H)
    (Finset.image
      (fun (g : G) ↦ Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g * (↑(hR.toRightFun (T := (↑R : Set G)) g) : G)⁻¹) sorry)
      (R * S)) ≤
  H.index * #S
Tactic: refine ⟨_, ?_, closure_mul_image_eq_top' hR hR1 hS⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
⊢ Finset.card (α := ↥H)
      (Finset.image
        (fun (g : G) ↦ Subtype.mk (p := fun (x : G) ↦ x ∈ H) (g * (↑(hR.toRightFun (T := (↑R : Set G)) g) : G)⁻¹) ⋯)
        (R * S)) ≤
    H.index * #S
have _ : #R = H.index
Tactic: calc
    _ ≤ #(R * S) := Finset.card_image_le
    _ ≤ #R * #S := Finset.card_mul_le
    _ = H.index * S.card := congr_arg (· * S.card) ?_
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
⊢ Fintype.card.{u_1} (G ⧸ H) = H.index
have rw : Fintype.card.{u_1} (G ⧸ H) = Nat.card.{u_1} (G ⧸ H)
Tactic: rw [index_eq_card, Nat.card_eq_fintype_card]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
rw : Fintype.card.{u_1} (G ⧸ H) = Nat.card.{u_1} (G ⧸ H)
⊢ Fintype.card.{u_1} (G ⧸ H) = H.index
have rw₁ : Fintype.card.{u_1} (G ⧸ H) = Fintype.card.{u_1} (G ⧸ H)
Tactic: rw [index_eq_card, Nat.card_eq_fintype_card]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
this_1 : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex
this_2 : DecidableEq G
R₀ : Set G
this : Fintype (↑R₀ : Type u_1)
R : Finset G := R₀.toFinset
hR : IsComplement (↑H : Set G) (↑R : Set G)
hR1 : 1 ∈ R
⊢ #R = H.index
have R₁ : Fintype.card.{u_1} (G ⧸ H) = H.index
Tactic: calc
    #R = Fintype.card R := (Fintype.card_coe R).symm
    _ = _ := (Fintype.card_congr hR.rightQuotientEquiv).symm
    _ = Fintype.card (G ⧸ H) := QuotientGroup.card_quotient_rightRel H
    _ = H.index := by rw [index_eq_card, Nat.card_eq_fintype_card]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
hG : Group.FG G
inst_1 : H.FiniteIndex
⊢ Group.FG.{u_1} ↥H
have obtain : ∀ (S : Finset G) (hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤), Group.FG.{u_1} ↥H
Tactic: obtain ⟨S, hS⟩ := hG.1
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S_1 : Set G
hG : Group.FG G
inst_1 : H.FiniteIndex
S : Finset G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
⊢ Group.FG.{u_1} ↥H
have obtain : ∀ (T : Finset ↥H) (hT : Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤), Group.FG.{u_1} ↥H
Tactic: obtain ⟨T, -, hT⟩ := exists_finset_card_le_mul H hS
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
hG : Group.FG G
inst_1 : H.FiniteIndex
⊢ Group.rank.{u_1} ↥H ≤ H.index * Group.rank G
have this : Group.FG.{u_1} ↥H
Tactic: haveI := H.fg_of_index_ne_zero
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
hG : Group.FG G
inst_1 : H.FiniteIndex
this : Group.FG.{u_1} ↥H
⊢ Group.rank.{u_1} ↥H ≤ H.index * Group.rank G
have obtain : ∀ (S : Finset G) (hS₀ : #S = Group.rank G) (hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤),
  Group.rank.{u_1} ↥H ≤ H.index * Group.rank G
Tactic: obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
hG : Group.FG G
inst_1 : H.FiniteIndex
this : Group.FG.{u_1} ↥H
S : Finset G
hS₀ : #S = Group.rank G
hS : Eq (α := Subgroup G) (closure (↑S : Set G)) ⊤
⊢ Group.rank.{u_1} ↥H ≤ H.index * Group.rank G
have obtain : ∀ (T : Finset ↥H) (hT₀ : #T ≤ H.index * #S) (hT : Eq (α := Subgroup ↥H) (closure (↑T : Set ↥H)) ⊤),
  Group.rank.{u_1} ↥H ≤ H.index * Group.rank G
Tactic: obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have pos : ∀ (hG : index (center G) = 0),
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
Tactic: by_cases hG : (center G).index = 0
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
pos :
  ∀ (hG : index (center G) = 0),
    Nat.card.{u_1} ↥(_root_.commutator G) ∣
      index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have neg : ∀ (hG : ¬index (center G) = 0),
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
Tactic: by_cases hG : (center G).index = 0
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : index (center G) = 0
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have pos : Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) 0 (0 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
Tactic: simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : index (center G) = 0
pos :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) 0 (0 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have pos₁ : Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) 0 (HAdd.hAdd (α := ℕ) 0 1)
Tactic: simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : index (center G) = 0
pos :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) 0 (0 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
pos₁ : Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) 0 (HAdd.hAdd (α := ℕ) 0 1)
pos₂ : Nat.card.{u_1} ↥(_root_.commutator G) ∣ HPow.hPow (α := ℕ) (β := ℕ) 0 1
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have pos₃ : Nat.card.{u_1} ↥(_root_.commutator G) ∣ 0
Tactic: simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have this : FiniteIndex (center G)
Tactic: haveI : FiniteIndex (center G) := ⟨hG⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)
have neg : Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) *
    index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) ∣
  index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)) * index (center G)
Tactic: rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ]
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) *
      index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)) * index (center G)
have h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
Tactic: have h1 := relIndex_dvd_index_of_normal (center G) (_root_.commutator G)
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) *
      index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)) * index (center G)
have neg : Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
  index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
Tactic: refine mul_dvd_mul ?_ h1
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have this₁ : FiniteIndex (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G))
Tactic: haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=
    ⟨ne_zero_of_dvd_ne_zero hG h1⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this_1 : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G))
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have h2 : Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ≤
  index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) *
    Group.rank.{u_1} ↥(_root_.commutator G)
Tactic: have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this_1 : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) *
      Group.rank.{u_1} ↥(_root_.commutator G)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have h3 : relIndex (center G) (_root_.commutator G) * Group.rank.{u_1} ↥(_root_.commutator G) ≤
  index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)
Tactic: have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this_1 : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) *
      Group.rank.{u_1} ↥(_root_.commutator G)
h3 :
  relIndex (center G) (_root_.commutator G) * Group.rank.{u_1} ↥(_root_.commutator G) ≤
    index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have neg : Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
  index (center G) ^ Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G))
Tactic: refine dvd_trans ?_ (pow_dvd_pow (center G).index (h2.trans h3))
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (goal)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
this_1 : FiniteIndex (center G)
h1 : relIndex (center G) (_root_.commutator G) ∣ index (center G)
this : FiniteIndex (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G))
h2 :
  Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ≤
    index (G := ↥(_root_.commutator G)) (subgroupOf (center G) (_root_.commutator G)) *
      Group.rank.{u_1} ↥(_root_.commutator G)
h3 :
  relIndex (center G) (_root_.commutator G) * Group.rank.{u_1} ↥(_root_.commutator G) ≤
    index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G))
have neg : ∀ (g : ↥(subgroupOf (center G) (_root_.commutator G))), g ^ index (center G) = 1
Tactic: apply card_dvd_exponent_pow_rank'
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have h1 : index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
  Nat.card.{u_1} (↑(commutatorSet ↥(closureCommutatorRepresentatives G)) : Type u_1) ^
    Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)
Tactic: have h1 := index_center_le_pow (closureCommutatorRepresentatives G)
-- Lean 4 proof step from Mathlib.GroupTheory.Schreier (hypothesis)
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
h1 :
  index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card.{u_1} (↑(commutatorSet ↥(closureCommutatorRepresentatives G)) : Type u_1) ^
      Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
have h2 : Nat.card.{u_1} ↥(_root_.commutator ↥(closureCommutatorRepresentatives G)) ∣
  index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
    (index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) *
        Nat.card.{u_1} (↑(commutatorSet ↥(closureCommutatorRepresentatives G)) : Type u_1) +
      1)
Tactic: have h2 := card_commutator_dvd_index_center_pow (closureCommutatorRepresentatives G)