l3lab/ntp-mathlib huggingface dataset:
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩
  · 

Proof state:
case refine'_3
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
y : Associates M
h : y ≤ p ^ n
⊢ ∃ i, y = (fun i => p ^ ↑i) i

Next tactic:
obtain ⟨i, i_le, hi⟩ := (dvd_prime_pow hp n).1 h
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩
  · obtain ⟨i, i_le, hi⟩ := (dvd_prime_pow hp n).1 h
    

Proof state:
case refine'_3.intro.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
y : Associates M
h : y ≤ p ^ n
i : ℕ
i_le : i ≤ n
hi : Associated y (p ^ i)
⊢ ∃ i, y = (fun i => p ^ ↑i) i

Next tactic:
rw [associated_iff_eq] at hi
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩
  · obtain ⟨i, i_le, hi⟩ := (dvd_prime_pow hp n).1 h
    rw [associated_iff_eq] at hi
    

Proof state:
case refine'_3.intro.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
y : Associates M
h : y ≤ p ^ n
i : ℕ
i_le : i ≤ n
hi : y = p ^ i
⊢ ∃ i, y = (fun i => p ^ ↑i) i

Next tactic:
exact ⟨⟨i, Nat.lt_succ_of_le i_le⟩, hi⟩
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩
  · obtain ⟨i, i_le, hi⟩ := (dvd_prime_pow hp n).1 h
    rw [associated_iff_eq] at hi
    exact ⟨⟨i, Nat.lt_succ_of_le i_le⟩, hi⟩
  · 

Proof state:
case refine'_4
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
y : Associates M
⊢ (∃ i, y = (fun i => p ^ ↑i) i) → y ≤ p ^ n

Next tactic:
rintro ⟨i, rfl⟩
theorem exists_chain_of_prime_pow {p : Associates M} {n : ℕ} (hn : n ≠ 0) (hp : Prime p) :
    ∃ c : Fin (n + 1) → Associates M,
      c 1 = p ∧ StrictMono c ∧ ∀ {r : Associates M}, r ≤ p ^ n ↔ ∃ i, r = c i := by
  refine' ⟨fun i => p ^ (i : ℕ), _, fun n m h => _, @fun y => ⟨fun h => _, _⟩⟩
  · dsimp only
    rw [Fin.val_one', Nat.mod_eq_of_lt, pow_one]
    exact Nat.lt_succ_of_le (Nat.one_le_iff_ne_zero.mpr hn)
  · exact Associates.dvdNotUnit_iff_lt.mp
        ⟨pow_ne_zero n hp.ne_zero, p ^ (m - n : ℕ),
          not_isUnit_of_not_isUnit_dvd hp.not_unit (dvd_pow dvd_rfl (Nat.sub_pos_of_lt h).ne'),
          (pow_mul_pow_sub p h.le).symm⟩
  · obtain ⟨i, i_le, hi⟩ := (dvd_prime_pow hp n).1 h
    rw [associated_iff_eq] at hi
    exact ⟨⟨i, Nat.lt_succ_of_le i_le⟩, hi⟩
  · rintro ⟨i, rfl⟩
    

Proof state:
case refine'_4.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : Associates M
n : ℕ
hn : n ≠ 0
hp : Prime p
i : Fin (n + 1)
⊢ (fun i => p ^ ↑i) i ≤ p ^ n

Next tactic:
exact ⟨p ^ (n - i : ℕ), (pow_mul_pow_sub p (Nat.succ_le_succ_iff.mp i.2)).symm⟩
theorem first_of_chain_isUnit {q : Associates M} {n : ℕ} {c : Fin (n + 1) → Associates M}
    (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : IsUnit (c 0) := by
  

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ IsUnit (c 0)

Next tactic:
obtain ⟨i, hr⟩ := h₂.mp Associates.one_le
theorem first_of_chain_isUnit {q : Associates M} {n : ℕ} {c : Fin (n + 1) → Associates M}
    (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : IsUnit (c 0) := by
  obtain ⟨i, hr⟩ := h₂.mp Associates.one_le
  

Proof state:
case intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1)
hr : 1 = c i
⊢ IsUnit (c 0)

Next tactic:
rw [Associates.isUnit_iff_eq_one, ← Associates.le_one_iff, hr]
theorem first_of_chain_isUnit {q : Associates M} {n : ℕ} {c : Fin (n + 1) → Associates M}
    (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i) : IsUnit (c 0) := by
  obtain ⟨i, hr⟩ := h₂.mp Associates.one_le
  rw [Associates.isUnit_iff_eq_one, ← Associates.le_one_iff, hr]
  

Proof state:
case intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
n : ℕ
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1)
hr : 1 = c i
⊢ c 0 ≤ c i

Next tactic:
exact h₁.monotone (Fin.zero_le i)
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero 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
⊢ Irreducible (c 1)

Next tactic:
cases' n with n
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · 

Proof state:
case zero
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
hn : Nat.zero ≠ 0
c : Fin (Nat.zero + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ Irreducible (c 1)

Next tactic:
contradiction
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  

Proof state:
case succ
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
⊢ Irreducible (c 1)

Next tactic:
refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
  · 

Proof state:
case succ.refine'_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
⊢ c 1 ≠ ⊥

Next tactic:
exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
  · exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
  

Proof state:
case succ.refine'_2
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
b : Associates M
hb : b < c 1
⊢ b = ⊥

Next tactic:
obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
  · exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
  obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
  

Proof state:
case succ.refine'_2.intro.mk
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
i : ℕ
hi : i < Nat.succ n + 1
hb : c { val := i, isLt := hi } < c 1
⊢ c { val := i, isLt := hi } = ⊥

Next tactic:
cases i
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
  · exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
  obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
  cases i
  · 

Proof state:
case succ.refine'_2.intro.mk.zero
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
hi : Nat.zero < Nat.succ n + 1
hb : c { val := Nat.zero, isLt := hi } < c 1
⊢ c { val := Nat.zero, isLt := hi } = ⊥

Next tactic:
exact (Associates.isUnit_iff_eq_one _).mp (first_of_chain_isUnit h₁ @h₂)
/-- The second element of a chain is irreducible. -/
theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
    {c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
    (hq : q ≠ 0) : Irreducible (c 1) := by
  cases' n with n; · contradiction
  refine' (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨_, fun b hb => _⟩
  · exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
  obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
  cases i
  · exact (Associates.isUnit_iff_eq_one _).mp (first_of_chain_isUnit h₁ @h₂)
  · 

Proof state:
case succ.refine'_2.intro.mk.succ
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q : Associates M
hq : q ≠ 0
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
n✝ : ℕ
hi : Nat.succ n✝ < Nat.succ n + 1
hb : c { val := Nat.succ n✝, isLt := hi } < c 1
⊢ c { val := Nat.succ n✝, isLt := hi } = ⊥

Next tactic:
simpa [Fin.lt_iff_val_lt_val] using h₁.lt_iff_lt.mp hb
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
  

Proof state:
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
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

Next tactic:
cases' n with n
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
  · 

Proof state:
case zero
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p q r : Associates M
hp : Prime p
hr : r ∣ q
hp' : p ∣ r
hn : Nat.zero ≠ 0
c : Fin (Nat.zero + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ p = c 1

Next tactic:
contradiction
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
  

Proof state:
case succ
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p q r : Associates M
hp : Prime p
hr : r ∣ q
hp' : p ∣ r
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
⊢ p = c 1

Next tactic:
obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
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)
  

Proof state:
case succ.intro
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
i : Fin (Nat.succ n + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ c i = c 1

Next tactic:
refine' congr_arg c (eq_of_ge_of_not_gt _ fun hi => _)
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 => _)
  · 

Proof state:
case succ.intro.refine'_1
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
i : Fin (Nat.succ n + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ 1 ≤ i

Next tactic:
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]
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]
    

Proof state:
case succ.intro.refine'_1
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
i : Fin (Nat.succ n + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ i ≠ 0

Next tactic:
rintro rfl
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
    

Proof state:
case succ.intro.refine'_1
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
hp : Prime (c 0)
hp' : c 0 ∣ r
⊢ False

Next tactic:
exact hp.not_unit (first_of_chain_isUnit h₁ @h₂)
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)
  

Proof state:
case succ.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
i : Fin (Nat.succ n + 1)
hp : Prime (c i)
hp' : c i ∣ r
hi : 1 < i
⊢ False

Next tactic:
obtain rfl | ⟨j, rfl⟩ := i.eq_zero_or_eq_succ
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
  · 

Proof state:
case succ.intro.refine'_2.inl
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
hp : Prime (c 0)
hp' : c 0 ∣ r
hi : 1 < 0
⊢ False

Next tactic:
cases hi
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
  

Proof state:
case succ.intro.refine'_2.inr.intro
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
⊢ False

Next tactic:
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
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
  · 

Proof state:
case succ.intro.refine'_2.inr.intro.refine'_1
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
⊢ 0 < ↑↑j

Next tactic:
simpa [Fin.succ_lt_succ_iff, Fin.lt_iff_val_lt_val] using hi
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
  · 

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
⊢ DvdNotUnit (c ↑↑j) (c (Fin.succ j))

Next tactic:
refine' Associates.dvdNotUnit_iff_lt.2 (h₁ _)