-- Module: Mathlib.RingTheory.Polynomial.Content
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
h : Nontrivial R
suffices suffices₁ suffices₂ suffices₃ :
  ∀ (n : ℕ) (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content
succ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑(n + 1) : WithBot ℕ)), (p * q).content = p.content * q.content
pos :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑(n + 1) : WithBot ℕ)) (p0 : p = 0), (p * q).content = p.content * q.content
neg neg₁ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑(n + 1) : WithBot ℕ)) (p0 : ¬p = 0), (p * q).content = p.content * q.content
pos₁ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑(n + 1) : WithBot ℕ)) (p0 : ¬p = 0) (q0 : q = 0),
    (p * q).content = p.content * q.content
neg₂ neg₃ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑(n + 1) : WithBot ℕ)) (p0 : ¬p = 0) (q0 : ¬q = 0),
    (p * q).content = p.content * q.content
neg₄ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ) ∨ p.natDegree + q.natDegree = n) (p0 : ¬p = 0) (q0 : ¬q = 0),
    (p * q).content = p.content * q.content
neg₅ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (hlt : (p * q).degree < (↑n : WithBot ℕ)),
    (p * q).content = p.content * q.content
neg₆ neg₇ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.natDegree + q.natDegree = n),
    (p * q).content = p.content * q.content
neg₈ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    (p * q).content = p.content * q.content
neg₉ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.content : R[X]) p.primPart *
          HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content =
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) p.content : R[X]) p.primPart).content *
        (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
neg₁₀ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    (q.primPart * p.primPart).content = 1
neg₁₁ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    @gcd _ CommSemiring.toCommMonoidWithZero _ (q.primPart * p.primPart).eraseLead.content
        (q.primPart.leadingCoeff * p.primPart.leadingCoeff) ∣
      1
neg₁₂ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    @gcd _ CommSemiring.toCommMonoidWithZero _ (q.primPart * p.primPart).eraseLead.content q.primPart.leadingCoeff *
        @gcd _ CommSemiring.toCommMonoidWithZero _ (q.primPart * p.primPart).eraseLead.content p.primPart.leadingCoeff ∣
      1
neg₁₃ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    (p.primPart.eraseLead * q.primPart).degree < (↑n : WithBot ℕ)
neg₁₄ neg₁₅ :
  ∀ (n : ℕ) (ih : ∀ (p q : R[X]) (hpq : (p * q).degree < (↑n : WithBot ℕ)), (p * q).content = p.content * q.content)
    (p q : R[X]) (p0 : ¬p = 0) (q0 : ¬q = 0) (heq : p.primPart.degree + q.primPart.degree = (↑n : WithBot ℕ)),
    (q.primPart.eraseLead * p.primPart).degree < (↑n : WithBot ℕ)
⊢ (p * q).content = p.content * q.content

Tactic:
classical
    suffices h :
        ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content by
      apply h
      apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
    intro n p q hpq
    induction n generalizing p q with
    | zero =>
      rw [Nat.cast_zero, Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
      rcases hpq with (rfl | rfl) <;> simp
    | succ n ih => ?_
    by_cases p0 : p = 0
    · simp [p0]
    by_cases q0 : q = 0
    · simp [q0]
    rw [degree_eq_natDegree (mul_ne_zero p0 q0), Nat.cast_lt,
      Nat.lt_succ_iff_lt_or_eq, ← Nat.cast_lt (α := WithBot ℕ),
      ← degree_eq_natDegree (mul_ne_zero p0 q0), natDegree_mul p0 q0] at hpq
    rcases hpq with (hlt | heq)
    · apply ih _ _ hlt
    rw [← p.natDegree_primPart, ← q.natDegree_primPart, ← Nat.cast_inj (R := WithBot ℕ),
      Nat.cast_add, ← degree_eq_natDegree p.primPart_ne_zero,
      ← degree_eq_natDegree q.primPart_ne_zero] at heq
    rw [p.eq_C_content_mul_primPart, q.eq_C_content_mul_primPart]
    suffices h : (q.primPart * p.primPart).content = 1 by
      rw [mul_assoc, content_C_mul, content_C_mul, mul_comm p.primPart, mul_assoc, content_C_mul,
        content_C_mul, h, mul_one, content_primPart, content_primPart, mul_one, mul_one]
    rw [← normalize_content, normalize_eq_one, isUnit_iff_dvd_one,
      content_eq_gcd_leadingCoeff_content_eraseLead, leadingCoeff_mul, gcd_comm]
    apply (gcd_mul_dvd_mul_gcd _ _ _).trans
    rw [content_mul_aux, ih, content_primPart, mul_one, gcd_comm, ←
      content_eq_gcd_leadingCoeff_content_eraseLead, content_primPart, one_mul,
      mul_comm q.primPart, content_mul_aux, ih, content_primPart, mul_one, gcd_comm, ←
      content_eq_gcd_leadingCoeff_content_eraseLead, content_primPart]
    · rw [← heq, degree_mul, WithBot.add_lt_add_iff_right]
      · apply degree_erase_lt p.primPart_ne_zero
      · rw [Ne, degree_eq_bot]
        apply q.primPart_ne_zero
    · rw [mul_comm, ← heq, degree_mul, WithBot.add_lt_add_iff_left]
      · apply degree_erase_lt q.primPart_ne_zero
      · rw [Ne, degree_eq_bot]
        apply p.primPart_ne_zero

Premises used: Polynomial.degree_eq_natDegree, Polynomial.leadingCoeff_mul, Polynomial.degree_erase_lt, Ne, Polynomial.eq_C_content_mul_primPart, Polynomial.normalize_content, Polynomial.natDegree_primPart, Polynomial.content_mul_aux, Nat.cast_inj, mul_eq_zero, Nat.WithBot.lt_zero_iff, gcd_mul_dvd_mul_gcd, gcd_comm, Nat.cast_add, Polynomial.primPart_ne_zero, Polynomial.primPart, mul_comm, Polynomial.content_primPart, Nat.lt_succ_iff_lt_or_eq, Nat.cast_zero, mul_one, Polynomial.content_eq_gcd_leadingCoeff_content_eraseLead, normalize_eq_one, Nat.cast_lt, isUnit_iff_dvd_one, one_mul, rfl, WithBot.add_lt_add_iff_left, Polynomial.degree_eq_bot, Dvd.dvd.trans, Polynomial.degree_mul, WithBot.add_lt_add_iff_right