l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  classical
    suffices h :
      ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content
    · apply h
      apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
    intro n
    induction' n with n ih
    · intro p q hpq
      dsimp at hpq
      rw [Nat.cast_zero,
        Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
      rcases hpq with (rfl | rfl) <;> simp
    intro p q hpq
    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
    · 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.def, degree_eq_bot]
        apply q.primPart_ne_zero
    · 

Proof state:
case h.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : degree (primPart p) + degree (primPart q) = ↑n
⊢ degree (eraseLead (primPart q) * primPart p) < ↑n

Next tactic:
rw [mul_comm, ← heq, degree_mul, WithBot.add_lt_add_iff_left]
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  classical
    suffices h :
      ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content
    · apply h
      apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
    intro n
    induction' n with n ih
    · intro p q hpq
      dsimp at hpq
      rw [Nat.cast_zero,
        Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
      rcases hpq with (rfl | rfl) <;> simp
    intro p q hpq
    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
    · 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.def, degree_eq_bot]
        apply q.primPart_ne_zero
    · rw [mul_comm, ← heq, degree_mul, WithBot.add_lt_add_iff_left]
      · 

Proof state:
case h.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : degree (primPart p) + degree (primPart q) = ↑n
⊢ degree (eraseLead (primPart q)) < degree (primPart q)

Next tactic:
apply degree_erase_lt q.primPart_ne_zero
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  classical
    suffices h :
      ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content
    · apply h
      apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
    intro n
    induction' n with n ih
    · intro p q hpq
      dsimp at hpq
      rw [Nat.cast_zero,
        Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
      rcases hpq with (rfl | rfl) <;> simp
    intro p q hpq
    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
    · 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.def, 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
      · 

Proof state:
case h.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : degree (primPart p) + degree (primPart q) = ↑n
⊢ degree (primPart p) ≠ ⊥

Next tactic:
rw [Ne.def, degree_eq_bot]
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  classical
    suffices h :
      ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content
    · apply h
      apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
    intro n
    induction' n with n ih
    · intro p q hpq
      dsimp at hpq
      rw [Nat.cast_zero,
        Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
      rcases hpq with (rfl | rfl) <;> simp
    intro p q hpq
    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
    · 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.def, 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.def, degree_eq_bot]
        

Proof state:
case h.a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ : R[X]
n : ℕ
ih : ∀ (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
p q : R[X]
p0 : ¬p = 0
q0 : ¬q = 0
heq : degree (primPart p) + degree (primPart q) = ↑n
⊢ ¬primPart p = 0

Next tactic:
apply p.primPart_ne_zero
theorem IsPrimitive.mul {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    (p * q).IsPrimitive := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
⊢ IsPrimitive (p * q)

Next tactic:
rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_one, mul_one]
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : p * q ≠ 0
⊢ primPart (p * q) = primPart p * primPart q

Next tactic:
rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (content (p * q)) = 0
⊢ primPart (p * q) = primPart p * primPart q

Next tactic:
apply mul_left_cancel₀ h0
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
  apply mul_left_cancel₀ h0
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (content (p * q)) = 0
⊢ C (content (p * q)) * primPart (p * q) = C (content (p * q)) * (primPart p * primPart q)

Next tactic:
conv_lhs =>
    rw [← (p * q).eq_C_content_mul_primPart, p.eq_C_content_mul_primPart,
      q.eq_C_content_mul_primPart]
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
  apply mul_left_cancel₀ h0
  conv_lhs =>
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (content (p * q)) = 0
| C (content (p * q)) * primPart (p * q)

Next tactic:
rw [← (p * q).eq_C_content_mul_primPart, p.eq_C_content_mul_primPart,
      q.eq_C_content_mul_primPart]
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
  apply mul_left_cancel₀ h0
  conv_lhs =>
    rw [← (p * q).eq_C_content_mul_primPart, p.eq_C_content_mul_primPart,
      q.eq_C_content_mul_primPart]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (content (p * q)) = 0
⊢ C (content p) * primPart p * (C (content q) * primPart q) = C (content (p * q)) * (primPart p * primPart q)

Next tactic:
rw [content_mul, RingHom.map_mul]
@[simp]
theorem primPart_mul {p q : R[X]} (h0 : p * q ≠ 0) :
    (p * q).primPart = p.primPart * q.primPart := by
  rw [Ne.def, ← content_eq_zero_iff, ← C_eq_zero] at h0
  apply mul_left_cancel₀ h0
  conv_lhs =>
    rw [← (p * q).eq_C_content_mul_primPart, p.eq_C_content_mul_primPart,
      q.eq_C_content_mul_primPart]
  rw [content_mul, RingHom.map_mul]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h0 : ¬C (content (p * q)) = 0
⊢ C (content p) * primPart p * (C (content q) * primPart q) = C (content p) * C (content q) * (primPart p * primPart q)

Next tactic:
ring
theorem IsPrimitive.dvd_primPart_iff_dvd {p q : R[X]} (hp : p.IsPrimitive) (hq : q ≠ 0) :
    p ∣ q.primPart ↔ p ∣ q := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : q ≠ 0
⊢ p ∣ primPart q ↔ p ∣ q

Next tactic:
refine' ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => _⟩
theorem IsPrimitive.dvd_primPart_iff_dvd {p q : R[X]} (hp : p.IsPrimitive) (hq : q ≠ 0) :
    p ∣ q.primPart ↔ p ∣ q := by
  refine' ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => _⟩
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : q ≠ 0
h : p ∣ q
⊢ p ∣ primPart q

Next tactic:
rcases h with ⟨r, rfl⟩
theorem IsPrimitive.dvd_primPart_iff_dvd {p q : R[X]} (hp : p.IsPrimitive) (hq : q ≠ 0) :
    p ∣ q.primPart ↔ p ∣ q := by
  refine' ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => _⟩
  rcases h with ⟨r, rfl⟩
  

Proof state:
case intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : IsPrimitive p
r : R[X]
hq : p * r ≠ 0
⊢ p ∣ primPart (p * r)

Next tactic:
apply Dvd.intro _
theorem IsPrimitive.dvd_primPart_iff_dvd {p q : R[X]} (hp : p.IsPrimitive) (hq : q ≠ 0) :
    p ∣ q.primPart ↔ p ∣ q := by
  refine' ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => _⟩
  rcases h with ⟨r, rfl⟩
  apply Dvd.intro _
  

Proof state:
case intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : IsPrimitive p
r : R[X]
hq : p * r ≠ 0
⊢ p * ?m.669400 = primPart (p * r)
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
hp : IsPrimitive p
r : R[X]
hq : p * r ≠ 0
⊢ R[X]

Next tactic:
rw [primPart_mul hq, hp.primPart_eq]
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s

Next 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
    · apply 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 (lt_of_not_ge 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)
    have hC0 := rprim.ne_zero
    rw [Ne.def, ← 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]
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s

Next tactic:
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 _ _⟩
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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 _ _⟩
    

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s

Next tactic:
rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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⟩
    

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : natDegree r = Nat.find h
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
⊢ ∃ r, IsPrimitive r ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s

Next tactic:
refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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⟩⟩⟩
    

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : natDegree r = Nat.find h
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
s : R[X]
⊢ p ∣ s ∧ q ∣ s → r ∣ s

Next tactic:
suffices hs : ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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
    · 

Proof state:
case intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : natDegree r = Nat.find h
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
s : R[X]
hs : ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s
⊢ p ∣ s ∧ q ∣ s → r ∣ s

Next tactic:
apply hs s.natDegree s rfl
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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
    · apply hs s.natDegree s rfl
    

Proof state:
case hs
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : natDegree r = Nat.find h
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
s : R[X]
⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s

Next tactic:
clear s
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  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
    · apply hs s.natDegree s rfl
    clear s
    

Proof state:
case hs
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : IsPrimitive p
hq : IsPrimitive q
h : ∃ n r, natDegree r = n ∧ IsPrimitive r ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : natDegree r = Nat.find h
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
⊢ ∀ (n : ℕ) (s : R[X]), natDegree s = n → p ∣ s ∧ q ∣ s → r ∣ s

Next tactic:
by_contra! con