awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.GroupTheory.Schreier
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
pos₃ : Nat.card.{u_1} ↥(_root_.commutator 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:
simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]

Premises used: dvd_zero, MulZeroClass.zero_mul, zero_add, pow_one
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
hG : ¬index (center G) = 0
⊢ FiniteIndex (center G)

Tactic:
haveI : FiniteIndex (center G) := ⟨hG⟩
-- Module: Mathlib.GroupTheory.Schreier
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)
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)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index (center G) ^ (index (center G) * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1)

Tactic:
rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ]

Premises used: Subgroup.subgroupOf, pow_succ, Subgroup.center, commutator, Subgroup.card_mul_index
-- Module: Mathlib.GroupTheory.Schreier
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)
⊢ relIndex (center G) (_root_.commutator G) ∣ index (center G)

Tactic:
have h1 := relIndex_dvd_index_of_normal (center G) (_root_.commutator G)
-- Module: Mathlib.GroupTheory.Schreier
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)

Tactic:
mul_dvd_mul ?_ h1

Premises used: mul_dvd_mul
-- Module: Mathlib.GroupTheory.Schreier
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)
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))
⊢ 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:
refine mul_dvd_mul ?_ h1

Premises used: mul_dvd_mul
-- Module: Mathlib.GroupTheory.Schreier
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)
⊢ 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⟩
-- Module: Mathlib.GroupTheory.Schreier
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))
⊢ 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))
-- Module: Mathlib.GroupTheory.Schreier
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)
⊢ 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)
-- Module: Mathlib.GroupTheory.Schreier
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))

Tactic:
dvd_trans ?_ (pow_dvd_pow (center G).index (h2.trans h3))

Premises used: pow_dvd_pow, Subgroup.center, dvd_trans, Subgroup.index, LE.le.trans
-- Module: Mathlib.GroupTheory.Schreier
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)
neg :
  Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ Group.rank.{u_1} ↥(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))

Tactic:
refine dvd_trans ?_ (pow_dvd_pow (center G).index (h2.trans h3))

Premises used: pow_dvd_pow, Subgroup.center, dvd_trans, Subgroup.index, LE.le.trans
-- Module: Mathlib.GroupTheory.Schreier
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)
neg : ∀ (g : ↥(subgroupOf (center G) (_root_.commutator G))), g ^ index (center G) = 1
⊢ Nat.card.{u_1} ↥(subgroupOf (center G) (_root_.commutator G)) ∣
    index (center G) ^ Group.rank.{u_1} ↥(subgroupOf (center G) (_root_.commutator G))

Tactic:
apply card_dvd_exponent_pow_rank'

Premises used: card_dvd_exponent_pow_rank'
-- Module: Mathlib.GroupTheory.Schreier
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_2 : 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)
g : ↥(subgroupOf (center G) (_root_.commutator G))
this :
  Membership.mem (γ := Subgroup G) (MonoidHom.ker (MonoidHom.transferCenterPow G)) (↑(↑g : ↥(_root_.commutator G)) : G)
⊢ g ^ index (center G) = 1

Tactic:
simpa only [MonoidHom.mem_ker, Subtype.ext_iff] using this

Premises used: MonoidHom.mem_ker, Subtype.ext_iff
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
⊢ 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)
-- Module: Mathlib.GroupTheory.Schreier
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 ↥(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)
-- Module: Mathlib.GroupTheory.Schreier
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)
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)
⊢ index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)

Tactic:
rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2

Premises used: card_commutatorSet_closureCommutatorRepresentatives
-- Module: Mathlib.GroupTheory.Schreier
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)
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)
h1₁ :
  index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)
⊢ 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 G) : Type u_1) +
        1)

Tactic:
rw [card_commutatorSet_closureCommutatorRepresentatives] at h1 h2

Premises used: card_commutatorSet_closureCommutatorRepresentatives
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)
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 G) : Type u_1) +
        1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) +
        1)

Tactic:
rw [card_commutator_closureCommutatorRepresentatives] at h2

Premises used: card_commutator_closureCommutatorRepresentatives
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ Group.rank.{u_1} ↥(closureCommutatorRepresentatives G)
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) +
        1)
⊢ index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))

Tactic:
replace h1 :=
    h1.trans
      (Nat.pow_le_pow_right Finite.card_pos (rank_closureCommutatorRepresentatives_le G))
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) *
          Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) +
        1)
h1 :
  index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ≤
    Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)) *
          Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) +
        1)

Tactic:
replace h2 := h2.trans (pow_dvd_pow _ (add_le_add_left (mul_le_mul_left h1 _) 1))
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1)) *
          Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) +
        1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)

Tactic:
rw [← pow_succ] at h2

Premises used: pow_succ
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))

Tactic:
(Nat.le_of_dvd ?_ h2).trans (Nat.pow_le_pow_left h1 _)

Premises used: LE.le.trans, Nat.le_of_dvd, Nat.pow_le_pow_left
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)
refine :
  0 <
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)
⊢ Nat.card.{u_1} ↥(_root_.commutator G) ≤ cardCommutatorBound (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))

Tactic:
refine (Nat.le_of_dvd ?_ h2).trans (Nat.pow_le_pow_left h1 _)

Premises used: LE.le.trans, Nat.le_of_dvd, Nat.pow_le_pow_left
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)
⊢ 0 <
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)

Tactic:
pow_pos (Nat.pos_of_ne_zero FiniteIndex.index_ne_zero) _

Premises used: pow_pos, Subgroup.FiniteIndex.index_ne_zero, Nat.pos_of_ne_zero
-- Module: Mathlib.GroupTheory.Schreier
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 G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1))
h2 :
  Nat.card.{u_1} ↥(_root_.commutator G) ∣
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)
⊢ 0 <
    index.{u_1} (G := ↥(closureCommutatorRepresentatives G)) (center ↥(closureCommutatorRepresentatives G)) ^
      (Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) ^ (2 * Nat.card.{u_1} (↑(commutatorSet G) : Type u_1) + 1) + 1)

Tactic:
exact pow_pos (Nat.pos_of_ne_zero FiniteIndex.index_ne_zero) _

Premises used: pow_pos, Subgroup.FiniteIndex.index_ne_zero, Nat.pos_of_ne_zero
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
⊢ 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)
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
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)
⊢ Finite.{u_1 + 1} ↥(_root_.commutator G)

Tactic:
Nat.finite_of_card_ne_zero fun h => ?_

Premises used: Nat.finite_of_card_ne_zero
-- Module: Mathlib.GroupTheory.Schreier
Goal:
G : Type u_1
inst : Group G
H : Subgroup G
R S : Set G
inst_1 : Finite.{u_1 + 1} (↑(commutatorSet G) : Type u_1)
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)
refine : ∀ (h : Nat.card.{u_1} ↥(_root_.commutator G) = 0), False
⊢ Finite.{u_1 + 1} ↥(_root_.commutator G)

Tactic:
refine Nat.finite_of_card_ne_zero fun h => ?_

Premises used: Nat.finite_of_card_ne_zero