-- Module: Mathlib.RingTheory.ChainOfDivisors
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
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
let₈ :
  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),
    (∃ (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a ∧ c 1 ^ (↑a : ℕ) = r) → r ≤ q
let₉ :
  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)
    (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a → c 1 ^ (↑a : ℕ) ≤ q
let₁₀ :
  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)
    (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a → c 1 ^ (↑a : ℕ) ∣ r
h :
  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)
    (a : Fin (i + 1)),
    Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a → r = c 1 ^ (↑a : ℕ) * c 1 ^ (i - (↑a : ℕ))
h₃ :
  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)
    (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a → r = c 1 ^ i
h₄ h₅ :
  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)
    (a : Fin (i + 1)), Membership.mem (γ := Finset (Fin (i + 1))) Finset.univ a → (↑a : ℕ) ≤ 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

Premises used: Finset.mem_image, dvd_trans, OfNat.ofNat, rfl, pow_mul_pow_sub