l3lab/ntp-mathlib huggingface dataset:
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 => 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
    

Proof state:
case H'.intro.intro
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
this : Finset.card (Finset.image (fun m => c 1 ^ ↑m) Finset.univ) = i + 1
a : Fin (Multiset.card (normalizedFactors r) + 1)
left✝ : a ∈ Finset.univ
⊢ c 1 ^ ↑a ∣ r

Next tactic:
use c 1 ^ (i - (a : ℕ))
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 => 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))
    

Proof state:
case h
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
this : Finset.card (Finset.image (fun m => c 1 ^ ↑m) Finset.univ) = i + 1
a : Fin (Multiset.card (normalizedFactors r) + 1)
left✝ : a ∈ Finset.univ
⊢ r = c 1 ^ ↑a * c 1 ^ (i - ↑a)

Next tactic:
rw [pow_mul_pow_sub (c 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 => 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)]
    · 

Proof state:
case h
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
this : Finset.card (Finset.image (fun m => c 1 ^ ↑m) Finset.univ) = i + 1
a : Fin (Multiset.card (normalizedFactors r) + 1)
left✝ : a ∈ Finset.univ
⊢ r = c 1 ^ i

Next tactic:
exact 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⟩
    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
    · 

Proof state:
case h
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
this : Finset.card (Finset.image (fun m => c 1 ^ ↑m) Finset.univ) = i + 1
a : Fin (Multiset.card (normalizedFactors r) + 1)
left✝ : a ∈ Finset.univ
⊢ ↑a ≤ i

Next tactic:
exact Nat.succ_le_succ_iff.mp a.2
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  

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

Next tactic:
classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    rcases h with ⟨u, hu, hu'⟩
    refine' Finset.mem_image.mpr ⟨u, Finset.mem_univ _, _⟩
    · rw [associated_iff_eq] at hu'
      rw [Fin.val_cast_of_lt (Nat.lt_succ_of_le hu), hu']
    · rw [← irreducible_iff_prime]
      exact second_of_chain_is_irreducible hn h₁ (@h₂) hq
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    

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

Next tactic:
obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    

Proof state:
case intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
⊢ q = c 1 ^ n

Next tactic:
convert hi'
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    

Proof state:
case h.e'_3.h.e'_6
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
⊢ n = ↑i

Next tactic:
refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    

Proof state:
case h.e'_3.h.e'_6
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
⊢ Nat.succ n ≤ Nat.succ ↑i

Next 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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    

Proof state:
case h.e'_3.h.e'_6
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
⊢ Finset.image c Finset.univ ⊆ Finset.image (fun m => c 1 ^ ↑m) Finset.univ

Next tactic:
intro r hr
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    

Proof state:
case h.e'_3.h.e'_6
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
r : Associates M
hr : r ∈ Finset.image c Finset.univ
⊢ r ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ

Next tactic:
obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    

Proof state:
case h.e'_3.h.e'_6.intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ

Next tactic:
have := h₂.2 ⟨j, rfl⟩
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    

Proof state:
case h.e'_3.h.e'_6.intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ q
⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ

Next tactic:
rw [hi'] at this
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    

Proof state:
case h.e'_3.h.e'_6.intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ ↑i
⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ

Next tactic:
have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    

Proof state:
case h.e'_3.h.e'_6.intro.intro.refine_2
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ ↑i
h : ∃ i_1 ≤ ↑i, Associated (c j) (c 1 ^ i_1)
⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ
case h.e'_3.h.e'_6.intro.intro.refine_1
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ ↑i
⊢ Prime (c 1)

Next tactic:
rcases h with ⟨u, hu, hu'⟩
theorem eq_pow_second_of_chain_of_has_chain {q : 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) (hq : q ≠ 0) : q = c 1 ^ n := by
  classical
    obtain ⟨i, hi'⟩ := element_of_chain_eq_pow_second_of_chain hn h₁ (@fun r => h₂) (dvd_refl q) hq
    convert hi'
    refine' (Nat.lt_succ_iff.1 i.prop).antisymm' (Nat.le_of_succ_le_succ _)
    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_of_subset ?_)
      _ ≤ (Finset.univ : Finset (Fin (i + 1))).card := Finset.card_image_le
      _ = i + 1 := Finset.card_fin _
    intro r hr
    obtain ⟨j, -, rfl⟩ := Finset.mem_image.1 hr
    have := h₂.2 ⟨j, rfl⟩
    rw [hi'] at this
    have h := (dvd_prime_pow (show Prime (c 1) from ?_) i).1 this
    rcases h with ⟨u, hu, hu'⟩
    

Proof state:
case h.e'_3.h.e'_6.intro.intro.refine_2.intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ ↑i
u : ℕ
hu : u ≤ ↑i
hu' : Associated (c j) (c 1 ^ u)
⊢ c j ∈ Finset.image (fun m => c 1 ^ ↑m) Finset.univ
case h.e'_3.h.e'_6.intro.intro.refine_1
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : 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, r = c i
hq : q ≠ 0
i : Fin (n + 1)
hi' : q = c 1 ^ ↑i
j : Fin (n + 1)
hr : c j ∈ Finset.image c Finset.univ
this : c j ≤ c 1 ^ ↑i
⊢ Prime (c 1)

Next tactic:
refine' Finset.mem_image.mpr ⟨u, Finset.mem_univ _, _⟩