-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
have have₁ have₂ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r),
    ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have₃ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r), ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have₄ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r) (s : R[X]), p ∣ s ∧ q ∣ s → r ∣ s
have₅ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r) (s : R[X]) (n : ℕ) (s : R[X]),
    s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s
have₆ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r) (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s
have₇ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r)
    (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s), False
have₈ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r)
    (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) (s : R[X])
    (sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con)
    (rs : ¬r ∣ s) (ps : p ∣ s) (qs : q ∣ s), False
have₉ :
  ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
    (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
    (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r)
    (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) (s : R[X])
    (sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con)
    (rs : ¬r ∣ s) (ps : p ∣ s) (qs : q ∣ s) (s0 : s ≠ 0), False
⊢ ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have have₁₀ : ∀ (h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) (r : R[X])
  (rdeg : r.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h)
  (rprim : r.IsPrimitive) (pr : p ∣ r) (qr : q ∣ r)
  (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) (s : R[X])
  (sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con)
  (rs : ¬r ∣ s) (ps : p ∣ s) (qs : q ∣ s) (s0 : s ≠ 0)
  (hs : Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h ≤ s.natDegree),
  False
Tactic: classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine ⟨r, rprim, fun s => ⟨?_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    suffices hs : ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s from
      hs s.natDegree s rfl
    clear s
    by_contra! con
    rcases Nat.find_spec con with ⟨s, sdeg, ⟨ps, qs⟩, rs⟩
    have s0 : s ≠ 0 := by
      contrapose rs
      simp [rs]
    have hs :=
      Nat.find_min' h
        ⟨_, s.natDegree_primPart, s.isPrimitive_primPart, (hp.dvd_primPart_iff_dvd s0).2 ps,
          (hq.dvd_primPart_iff_dvd s0).2 qs⟩
    rw [← rdeg] at hs
    by_cases! sC : s.natDegree ≤ 0
    · rw [eq_C_of_natDegree_le_zero (le_trans hs sC), isPrimitive_iff_content_eq_one, content_C,
        normalize_eq_one] at rprim
      rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs
      apply rs rprim.dvd
    have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs sC
    rw [sdeg] at hcancel
    apply Nat.find_min con hcancel
    refine
      ⟨_, rfl, ⟨dvd_cancelLeads_of_dvd_of_dvd pr ps, dvd_cancelLeads_of_dvd_of_dvd qr qs⟩,
        fun rcs => rs ?_⟩
    rw [← rprim.dvd_primPart_iff_dvd s0]
    rw [cancelLeads, tsub_eq_zero_iff_le.mpr hs, pow_zero, mul_one] at rcs
    have h :=
      dvd_add rcs (Dvd.intro_left (C (leadingCoeff s) * X ^ (natDegree s - natDegree r)) rfl)
    nontriviality R
    have hC0 := rprim.ne_zero
    rw [Ne, ← leadingCoeff_eq_zero, ← C_eq_zero] at hC0
    rw [sub_add_cancel, ← rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)] at h
    rcases isUnit_primPart_C r.leadingCoeff with ⟨u, hu⟩
    apply h.trans (Associated.symm ⟨u, _⟩).dvd
    rw [primPart_mul (mul_ne_zero hC0 s0), hu, mul_comm]