l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by 

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
⊢ g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H

Next tactic:
rwa [← zpow_ofNat, ← hk₄]
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
hk₅ : g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H
h : ¬Int.natAbs (k % ↑(Nat.find hex)) = 0
⊢ ↑(Int.natAbs (k % ↑(Nat.find hex))) < ↑(Nat.find hex)

Next tactic:
rw [← hk₄]
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; 

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
hk₅ : g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H
h : ¬Int.natAbs (k % ↑(Nat.find hex)) = 0
⊢ k % ↑(Nat.find hex) < ↑(Nat.find hex)

Next tactic:
exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1)
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
hk₅ : g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H
hk₆ : Int.natAbs (k % ↑(Nat.find hex)) = 0
⊢ ↑((fun x => { val := g ^ Nat.find hex, property := (_ : g ^ Nat.find hex ∈ H) } ^ x) (k / ↑(Nat.find hex))) =
    ↑{ val := x, property := hx }

Next tactic:
suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by simpa [zpow_mul]
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by 

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this✝ : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
hk₅ : g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H
hk₆ : Int.natAbs (k % ↑(Nat.find hex)) = 0
this : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) = x
⊢ ↑((fun x => { val := g ^ Nat.find hex, property := (_ : g ^ Nat.find hex ∈ H) } ^ x) (k / ↑(Nat.find hex))) =
    ↑{ val := x, property := hx }

Next tactic:
simpa [zpow_mul]
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by simpa [zpow_mul]
              

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx✝ : ∃ x ∈ H, x ≠ 1
x✝¹ : α
hx₁ : x✝¹ ∈ H
hx₂ : x✝¹ ≠ 1
k✝ : ℤ
hk✝² : (fun x => g ^ x) k✝ = x✝¹
hk✝¹ : g ^ k✝ = x✝¹
hex : ∃ n, 0 < n ∧ g ^ n ∈ H
x✝ : ↥H
x : α
hx : x ∈ H
k : ℤ
hk✝ : (fun x => g ^ x) k = x
hk : g ^ k = x
hk₂ : g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) ∈ H
hk₃ : g ^ (k % ↑(Nat.find hex)) ∈ H
hk₄ : k % ↑(Nat.find hex) = ↑(Int.natAbs (k % ↑(Nat.find hex)))
hk₅ : g ^ Int.natAbs (k % ↑(Nat.find hex)) ∈ H
hk₆ : Int.natAbs (k % ↑(Nat.find hex)) = 0
⊢ g ^ (↑(Nat.find hex) * (k / ↑(Nat.find hex))) = x

Next tactic:
rw [Int.mul_ediv_cancel'
                  (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero.mp hk₆)),
                hk]
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by simpa [zpow_mul]
              rw [Int.mul_ediv_cancel'
                  (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero.mp hk₆)),
                hk])⟩⟩⟩
  else by
    

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx : ¬∃ x ∈ H, x ≠ 1
⊢ IsCyclic ↥H

Next tactic:
have : H = (⊥ : Subgroup α) :=
      Subgroup.ext fun x =>
        ⟨fun h => by simp at *; tauto, fun h => by rw [Subgroup.mem_bot.1 h]; exact H.one_mem⟩
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by simpa [zpow_mul]
              rw [Int.mul_ediv_cancel'
                  (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero.mp hk₆)),
                hk])⟩⟩⟩
  else by
    have : H = (⊥ : Subgroup α) :=
      Subgroup.ext fun x =>
        ⟨fun h => by 

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
hx : ¬∃ x ∈ H, x ≠ 1
x : α
h : x ∈ H
⊢ x ∈ ⊥

Next tactic:
simp at *
@[to_additive]
instance Subgroup.isCyclic {α : Type u} [Group α] [IsCyclic α] (H : Subgroup α) : IsCyclic H :=
  haveI := Classical.propDecidable
  let ⟨g, hg⟩ := IsCyclic.exists_generator (α := α)
  if hx : ∃ x : α, x ∈ H ∧ x ≠ (1 : α) then
    let ⟨x, hx₁, hx₂⟩ := hx
    let ⟨k, hk⟩ := hg x
    have hk : g ^ k = x := hk
    have hex : ∃ n : ℕ, 0 < n ∧ g ^ n ∈ H :=
      ⟨k.natAbs,
        Nat.pos_of_ne_zero fun h => hx₂ <| by
          rw [← hk, Int.natAbs_eq_zero.mp h, zpow_zero], by
            cases' k with k k
            · rw [Int.ofNat_eq_coe, Int.natAbs_cast k, ← zpow_ofNat, ← Int.ofNat_eq_coe, hk]
              exact hx₁
            · rw [Int.natAbs_negSucc, ← Subgroup.inv_mem_iff H]; simp_all⟩
    ⟨⟨⟨g ^ Nat.find hex, (Nat.find_spec hex).2⟩, fun ⟨x, hx⟩ =>
        let ⟨k, hk⟩ := hg x
        have hk : g ^ k = x := hk
        have hk₂ : g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) ∈ H := by
          rw [zpow_mul]
          apply H.zpow_mem
          exact mod_cast (Nat.find_spec hex).2
        have hk₃ : g ^ (k % Nat.find hex : ℤ) ∈ H :=
          (Subgroup.mul_mem_cancel_right H hk₂).1 <| by
            rw [← zpow_add, Int.emod_add_ediv, hk]; exact hx
        have hk₄ : k % Nat.find hex = (k % Nat.find hex).natAbs := by
          rw [Int.natAbs_of_nonneg
              (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (Nat.find_spec hex).1))]
        have hk₅ : g ^ (k % Nat.find hex).natAbs ∈ H := by rwa [← zpow_ofNat, ← hk₄]
        have hk₆ : (k % (Nat.find hex : ℤ)).natAbs = 0 :=
          by_contradiction fun h =>
            Nat.find_min hex
              (Int.ofNat_lt.1 <| by
                rw [← hk₄]; exact Int.emod_lt_of_pos _ (Int.coe_nat_pos.2 (Nat.find_spec hex).1))
              ⟨Nat.pos_of_ne_zero h, hk₅⟩
        ⟨k / (Nat.find hex : ℤ),
          Subtype.ext_iff_val.2
            (by
              suffices g ^ ((Nat.find hex : ℤ) * (k / Nat.find hex : ℤ)) = x by simpa [zpow_mul]
              rw [Int.mul_ediv_cancel'
                  (Int.dvd_of_emod_eq_zero (Int.natAbs_eq_zero.mp hk₆)),
                hk])⟩⟩⟩
  else by
    have : H = (⊥ : Subgroup α) :=
      Subgroup.ext fun x =>
        ⟨fun h => by simp at *; 

Proof state:
α✝ : Type u
a : α✝
inst✝² : Group α✝
α : Type u
inst✝¹ : Group α
inst✝ : IsCyclic α
H : Subgroup α
this : (a : Prop) → Decidable a
g : α
hg : ∀ (x : α), x ∈ zpowers g
x : α
h : x ∈ H
hx : ∀ x ∈ H, x = 1
⊢ x = 1

Next tactic:
tauto