l3lab/ntp-mathlib huggingface dataset:
theorem eq_second_of_chain_of_prime_dvd {p 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, r = c i) (hp : Prime p) (hr : r ∣ q) (hp' : p ∣ r) :
    p = c 1 := by
  cases' n with n
  · contradiction
  obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
  refine' congr_arg c (eq_of_ge_of_not_gt _ fun hi => _)
  · rw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, ← Fin.val_zero' (n.succ + 1), ←
      Fin.lt_iff_val_lt_val, Fin.pos_iff_ne_zero]
    rintro rfl
    exact hp.not_unit (first_of_chain_isUnit h₁ @h₂)
  obtain rfl | ⟨j, rfl⟩ := i.eq_zero_or_eq_succ
  · cases hi
  refine'
    not_irreducible_of_not_unit_dvdNotUnit
      (DvdNotUnit.not_unit
        (Associates.dvdNotUnit_iff_lt.2 (h₁ (show (0 : Fin (n + 2)) < j from _))))
      _ hp.irreducible
  · simpa [Fin.succ_lt_succ_iff, Fin.lt_iff_val_lt_val] using hi
  · refine' Associates.dvdNotUnit_iff_lt.2 (h₁ _)
    

Proof state:
case succ.intro.refine'_2.inr.intro.refine'_2
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : Nat.succ n ≠ 0
c : Fin (Nat.succ n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c (Fin.succ j))
hp' : c (Fin.succ j) ∣ r
hi : 1 < Fin.succ j
⊢ ↑↑j < Fin.succ j

Next tactic:
simpa only [Fin.coe_eq_castSucc] using Fin.lt_succ
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ Finset.card m ≤ n + 1

Next tactic:
classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    rw [← Finset.card_fin (n + 1)]
    exact (Finset.card_le_of_subset fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ Finset.card m ≤ n + 1

Next tactic:
have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
⊢ ∀ r ≤ q, r ∈ Finset.image c Finset.univ

Next tactic:
intro r hr
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
r : Associates M
hr : r ≤ q
⊢ r ∈ Finset.image c Finset.univ

Next tactic:
obtain ⟨i, hi⟩ := h₂.1 hr
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      

Proof state:
case intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
r : Associates M
hr : r ≤ q
i : Fin (n + 1)
hi : r = c i
⊢ r ∈ Finset.image c Finset.univ

Next tactic:
exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
⊢ Finset.card m ≤ n + 1

Next tactic:
rw [← Finset.card_fin (n + 1)]
theorem card_subset_divisors_le_length_of_chain {q : Associates M} {n : ℕ}
    {c : Fin (n + 1) → Associates M} (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) {m : Finset (Associates M)}
    (hm : ∀ r, r ∈ m → r ≤ q) : m.card ≤ n + 1 := by
  classical
    have mem_image : ∀ r : Associates M, r ≤ q → r ∈ Finset.univ.image c := by
      intro r hr
      obtain ⟨i, hi⟩ := h₂.1 hr
      exact Finset.mem_image.2 ⟨i, Finset.mem_univ _, hi.symm⟩
    rw [← Finset.card_fin (n + 1)]
    

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
m : Finset (Associates M)
hm : ∀ r ∈ m, r ≤ q
mem_image : ∀ r ≤ q, r ∈ Finset.image c Finset.univ
⊢ Finset.card m ≤ Finset.card Finset.univ

Next tactic:
exact (Finset.card_le_of_subset fun x hx => mem_image x <| hm x hx).trans Finset.card_image_le
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
⊢ ∃ i, r = c 1 ^ ↑i

Next 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.normalizedFactors_prod (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_unit (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
    · 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
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  classical
    

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
⊢ ∃ i, r = c 1 ^ ↑i

Next tactic:
let i := Multiset.card (normalizedFactors r)
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  classical
    let i := Multiset.card (normalizedFactors r)
    

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
⊢ ∃ i, r = c 1 ^ ↑i

Next tactic:
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)
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
⊢ normalizedFactors r = Multiset.replicate i (c 1)

Next tactic:
apply Multiset.eq_replicate_of_mem
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  classical
    let i := Multiset.card (normalizedFactors r)
    have hi : normalizedFactors r = Multiset.replicate i (c 1) := by
      apply Multiset.eq_replicate_of_mem
      

Proof state:
case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
⊢ ∀ b ∈ normalizedFactors r, b = c 1

Next tactic:
intro b hb
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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
      

Proof state:
case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
b : Associates M
hb : b ∈ normalizedFactors r
⊢ b = c 1

Next tactic:
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)
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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)
    

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
⊢ ∃ i, r = c 1 ^ ↑i

Next tactic:
have H : r = c 1 ^ i := by
      have := UniqueFactorizationMonoid.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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
      

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
⊢ r = c 1 ^ i

Next tactic:
have := UniqueFactorizationMonoid.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
      

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
this : Associated (Multiset.prod (normalizedFactors r)) r
⊢ r = c 1 ^ i

Next tactic:
rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
this : c 1 ^ i = r
⊢ r = c 1 ^ i

Next tactic:
rw [this]
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ ∃ i, r = c 1 ^ ↑i

Next tactic:
refine' ⟨⟨i, _⟩, H⟩
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (ne_zero_of_dvd_ne_zero hq hr)
      rw [associated_iff_eq, hi, Multiset.prod_replicate] at this
      rw [this]
    refine' ⟨⟨i, _⟩, H⟩
    

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ i < n + 1

Next 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_unit (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)
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (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
      

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
⊢ Finset.card (Finset.image (fun m => c 1 ^ ↑m) Finset.univ) = i + 1

Next tactic:
conv_rhs => rw [← Finset.card_fin (i + 1)]
theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) := by
  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.normalizedFactors_prod (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 => 

Proof state:
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := Multiset.card (normalizedFactors r)
hi : normalizedFactors r = Multiset.replicate i (c 1)
H : r = c 1 ^ i
| i + 1

Next tactic:
rw [← Finset.card_fin (i + 1)]