awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ i < n + 1
have this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
Tactic: have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
H' :
  ∀ (r : Associates M),
    Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r →
      r ≤ q
⊢ i < n + 1
have simp : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) ≤ n + 1
Tactic: simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
⊢ i < n + 1
have suffices : ∀ (r : Associates M),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r →
    r ≤ q
Tactic: suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
⊢ ∀ (r : Associates M),
    Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r →
      r ≤ q
have simp : ∀ (r : Associates M),
  (∃ (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a ∧ c 1 ^ (↑a : ℕ) = r) → r ≤ q
Tactic: simp only [Finset.mem_image]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
a : Fin (i + 1)
left : Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a
⊢ c 1 ^ (↑a : ℕ) ≤ q
have refine : c 1 ^ (↑a : ℕ) ∣ r
Tactic: refine dvd_trans ?_ hr
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
a : Fin (i + 1)
left : Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a
⊢ c 1 ^ (↑a : ℕ) ∣ r
have h : r = c 1 ^ (↑a : ℕ) * c 1 ^ (i - (↑a : ℕ))
Tactic: use c 1 ^ (i - (a : ℕ))
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
this : Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1
a : Fin (i + 1)
left : Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a
h : r = c 1 ^ i
⊢ r = c 1 ^ (↑a : ℕ) * c 1 ^ (i - (↑a : ℕ))
have h₃ : (↑a : ℕ) ≤ i
Tactic: rw [pow_mul_pow_sub (c 1)]
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
let let₁ let₂ : ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have let₃ : let i := (normalizedFactors r).card;
∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
Tactic: classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
    have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine ⟨⟨i, ?_⟩, H⟩
    have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
    suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
    simp only [Finset.mem_image]
    rintro r ⟨a, _, rfl⟩
    refine dvd_trans ?_ hr
    use c 1 ^ (i - (a : ℕ))
    rw [pow_mul_pow_sub (c 1)]
    · exact H
    · exact Nat.succ_le_succ_iff.mp a.2
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
let let₁ let₂ : ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₃ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have let₄ : let i := (normalizedFactors r).card;
∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
Tactic: classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
    have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine ⟨⟨i, ?_⟩, H⟩
    have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
    suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
    simp only [Finset.mem_image]
    rintro r ⟨a, _, rfl⟩
    refine dvd_trans ?_ hr
    use c 1 ^ (i - (a : ℕ))
    rw [pow_mul_pow_sub (c 1)]
    · exact H
    · exact Nat.succ_le_succ_iff.mp a.2
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
let let₁ let₂ : ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₃ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₄ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have let₅ : let i := (normalizedFactors r).card;
∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), i < n + 1
Tactic: classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
    have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine ⟨⟨i, ?_⟩, H⟩
    have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
    suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
    simp only [Finset.mem_image]
    rintro r ⟨a, _, rfl⟩
    refine dvd_trans ?_ hr
    use c 1 ^ (i - (a : ℕ))
    rw [pow_mul_pow_sub (c 1)]
    · exact H
    · exact Nat.succ_le_succ_iff.mp a.2
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
let let₁ let₂ : ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₃ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₄ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₅ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), i < n + 1
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have let₆ : let i := (normalizedFactors r).card;
∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i)
  (this :
    Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1),
  i < n + 1
Tactic: classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
    have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine ⟨⟨i, ?_⟩, H⟩
    have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
    suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
    simp only [Finset.mem_image]
    rintro r ⟨a, _, rfl⟩
    refine dvd_trans ?_ hr
    use c 1 ^ (i - (a : ℕ))
    rw [pow_mul_pow_sub (c 1)]
    · exact H
    · exact Nat.succ_le_succ_iff.mp a.2
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hr : r ∣ q
hq : q ≠ 0
let let₁ let₂ : ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₃ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₄ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
let₅ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i), i < n + 1
let₆ :
  let i := (normalizedFactors r).card;
  ∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i)
    (this :
      Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) =
        i + 1),
    i < n + 1
⊢ ∃ (i : Fin (n + 1)), r = c 1 ^ (↑i : ℕ)
have let₇ : let i := (normalizedFactors r).card;
∀ (hi : normalizedFactors r = Multiset.replicate i (c 1)) (H : r = c 1 ^ i)
  (this :
    Finset.card.{u_1} (α := Associates M) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) = i + 1)
  (r : Associates M),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin (i + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) r →
    r ≤ q
Tactic: classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      intro b hb
      refine
        eq_second_of_chain_of_prime_dvd hn h₁ (@fun r' => h₂) (prime_of_normalized_factor b hb) hr
          (dvd_of_mem_normalizedFactors hb)
    have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.prod_normalizedFactors (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine ⟨⟨i, ?_⟩, H⟩
    have : (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card = i + 1 := by
      conv_rhs => rw [← Finset.card_fin (i + 1)]
      cases n
      · contradiction
      rw [Finset.card_image_iff]
      refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
      refine
        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
      exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
    suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
      simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
      apply card_subset_divisors_le_length_of_chain (@h₂) H'
    simp only [Finset.mem_image]
    rintro r ⟨a, _, rfl⟩
    refine dvd_trans ?_ hr
    use c 1 ^ (i - (a : ℕ))
    rw [pow_mul_pow_sub (c 1)]
    · exact H
    · exact Nat.succ_le_succ_iff.mp a.2
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
⊢ q = c 1 ^ n
have obtain : ∀ (i : Fin (n + 1)) (hi' : q = c 1 ^ (↑i : ℕ)), q = c 1 ^ n
Tactic: obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
⊢ q = c 1 ^ n
have h : n = (↑i : ℕ)
Tactic: convert! hi'
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
⊢ n = (↑i : ℕ)
have h : n.succ ≤ (↑i : ℕ).succ
Tactic: refine (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ ?_)
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
⊢ n.succ ≤ (↑i : ℕ).succ
have h : Finset.image c Finset.univ ⊆ Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ
Tactic: calc
      n + 1 = (Finset.univ : Finset (Fin (n + 1))).card := (Finset.card_fin _).symm
      _ = (Finset.univ.image c).card := (Finset.card_image_iff.mpr h₁.injective.injOn).symm
      _ ≤ (Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ)).card :=
        (Finset.card_le_card ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
r : Associates M
hr : r ∈ Finset.image c Finset.univ
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    r
have h : ∀ (j : Fin (n + 1)) (hr : c j ∈ Finset.image c Finset.univ),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
Cleared hypotheses: r, hr
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have this : c j ≤ q
Tactic: have := h₂.2 ⟨j, rfl⟩
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (hypothesis)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ q
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have this₁ : c j ≤ c 1 ^ (↑i : ℕ)
Tactic: rw [hi'] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have h : ∀ (h : ∃ i_1 ≤ (↑i : ℕ), Associated (c j) (c 1 ^ i_1)),
  Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
Tactic: have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
-- Lean 4 proof step from Mathlib.RingTheory.ChainOfDivisors (goal)
Goal:
M : Type u_1
inst : CommMonoidWithZero M
inst_1 : UniqueFactorizationMonoid M
q : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ (i : Fin (n + 1)), r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ (↑i : ℕ)
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ (↑i : ℕ)
h :
  ∀ (h : ∃ i_1 ≤ (↑i : ℕ), Associated (c j) (c 1 ^ i_1)),
    Membership.mem (γ := Finset (Associates M))
      (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ) (c j)
⊢ Membership.mem (γ := Finset (Associates M)) (Finset.image (fun (m : Fin ((↑i : ℕ) + 1)) ↦ c 1 ^ (↑m : ℕ)) Finset.univ)
    (c j)
have h₃ : Prime (c 1)
Tactic: have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this