-- Mathlib/RingTheory/ChainOfDivisors.lean
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

Proof state:
case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 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

Tactic:
exact hp.not_unit (first_of_chain_isUnit h₁ @h₂)

Result:
no goals

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 : n + 1 ≠ 0
c : Fin (n + 1 + 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

Tactic:
cases hi

Result:
no goals

Proof state:
case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ 1 ≤ i

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]

Result:
case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ i ≠ 0

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ False

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

Result:
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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ 0 < ↑↑j

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ DvdNotUnit (c ↑↑j) (c j.succ)

Proof state:
case succ.intro.refine_2
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
hi : 1 < i
⊢ False

Tactic:
obtain rfl | ⟨j, rfl⟩ := i.eq_zero_or_eq_succ

Result:
case succ.intro.refine_2.inl
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 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

case succ.intro.refine_2.inr.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ False

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ p = c 1

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

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

Proof state:
case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
i : Fin (n + 1 + 1)
hp : Prime (c i)
hp' : c i ∣ r
⊢ i ≠ 0

Tactic:
rintro rfl

Result:
case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 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

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 : 0 ≠ 0
c : Fin (0 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ p = c 1

Tactic:
contradiction

Result:
no goals

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ 0 < ↑↑j

Tactic:
simpa using Fin.lt_def.mp hi

Result:
no goals

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ ↑↑j < j.succ

Tactic:
simpa only [Fin.coe_eq_castSucc] using Fin.lt_succ

Result:
no goals

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ DvdNotUnit (c ↑↑j) (c j.succ)

Tactic:
refine Associates.dvdNotUnit_iff_lt.2 (h₁ ?_)

Result:
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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
j : Fin (n + 1)
hp : Prime (c j.succ)
hp' : c j.succ ∣ r
hi : 1 < j.succ
⊢ ↑↑j < j.succ

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

Tactic:
rcases n with - | n

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

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 : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
⊢ p = c 1