l3lab/ntp-mathlib huggingface dataset:
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
h : C a ∣ p - q
⊢ gcd a (content p) = gcd a (content q)

Next tactic:
rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
h : C a ∣ p - q
⊢ gcd a (Finset.gcd (Finset.range (Nat.succ (max (natDegree p) (natDegree q)))) (coeff p)) = gcd a (content q)

Next tactic:
rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
h : C a ∣ p - q
⊢ gcd a (Finset.gcd (Finset.range (Nat.succ (max (natDegree p) (natDegree q)))) (coeff p)) =
    gcd a (Finset.gcd (Finset.range (Nat.succ (max (natDegree p) (natDegree q)))) (coeff q))

Next tactic:
apply Finset.gcd_eq_of_dvd_sub
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
  apply Finset.gcd_eq_of_dvd_sub
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
h : C a ∣ p - q
⊢ ∀ x ∈ Finset.range (Nat.succ (max (natDegree p) (natDegree q))), a ∣ coeff p x - coeff q x

Next tactic:
intro x _
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
  apply Finset.gcd_eq_of_dvd_sub
  intro x _
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
h : C a ∣ p - q
x : ℕ
a✝ : x ∈ Finset.range (Nat.succ (max (natDegree p) (natDegree q)))
⊢ a ∣ coeff p x - coeff q x

Next tactic:
cases' h with w hw
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
  apply Finset.gcd_eq_of_dvd_sub
  intro x _
  cases' h with w hw
  

Proof state:
case h.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
x : ℕ
a✝ : x ∈ Finset.range (Nat.succ (max (natDegree p) (natDegree q)))
w : R[X]
hw : p - q = C a * w
⊢ a ∣ coeff p x - coeff q x

Next tactic:
use w.coeff x
theorem gcd_content_eq_of_dvd_sub {a : R} {p q : R[X]} (h : C a ∣ p - q) :
    GCDMonoid.gcd a p.content = GCDMonoid.gcd a q.content := by
  rw [content_eq_gcd_range_of_lt p (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_left _ _) (Nat.lt_succ_self _))]
  rw [content_eq_gcd_range_of_lt q (max p.natDegree q.natDegree).succ
      (lt_of_le_of_lt (le_max_right _ _) (Nat.lt_succ_self _))]
  apply Finset.gcd_eq_of_dvd_sub
  intro x _
  cases' h with w hw
  use w.coeff x
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
a : R
p q : R[X]
x : ℕ
a✝ : x ∈ Finset.range (Nat.succ (max (natDegree p) (natDegree q)))
w : R[X]
hw : p - q = C a * w
⊢ coeff p x - coeff q x = a * coeff w x

Next tactic:
rw [← coeff_sub, hw, coeff_C_mul]
theorem content_mul_aux {p q : R[X]} :
    GCDMonoid.gcd (p * q).eraseLead.content p.leadingCoeff =
      GCDMonoid.gcd (p.eraseLead * q).content p.leadingCoeff := by
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ gcd (content (eraseLead (p * q))) (leadingCoeff p) = gcd (content (eraseLead p * q)) (leadingCoeff p)

Next tactic:
rw [gcd_comm (content _) _, gcd_comm (content _) _]
theorem content_mul_aux {p q : R[X]} :
    GCDMonoid.gcd (p * q).eraseLead.content p.leadingCoeff =
      GCDMonoid.gcd (p.eraseLead * q).content p.leadingCoeff := by
  rw [gcd_comm (content _) _, gcd_comm (content _) _]
  

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ gcd (leadingCoeff p) (content (eraseLead (p * q))) = gcd (leadingCoeff p) (content (eraseLead p * q))

Next tactic:
apply gcd_content_eq_of_dvd_sub
theorem content_mul_aux {p q : R[X]} :
    GCDMonoid.gcd (p * q).eraseLead.content p.leadingCoeff =
      GCDMonoid.gcd (p.eraseLead * q).content p.leadingCoeff := by
  rw [gcd_comm (content _) _, gcd_comm (content _) _]
  apply gcd_content_eq_of_dvd_sub
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ C (leadingCoeff p) ∣ eraseLead (p * q) - eraseLead p * q

Next tactic:
rw [← self_sub_C_mul_X_pow, ← self_sub_C_mul_X_pow, sub_mul, sub_sub, add_comm, sub_add,
    sub_sub_cancel, leadingCoeff_mul, RingHom.map_mul, mul_assoc, mul_assoc]
theorem content_mul_aux {p q : R[X]} :
    GCDMonoid.gcd (p * q).eraseLead.content p.leadingCoeff =
      GCDMonoid.gcd (p.eraseLead * q).content p.leadingCoeff := by
  rw [gcd_comm (content _) _, gcd_comm (content _) _]
  apply gcd_content_eq_of_dvd_sub
  rw [← self_sub_C_mul_X_pow, ← self_sub_C_mul_X_pow, sub_mul, sub_sub, add_comm, sub_add,
    sub_sub_cancel, leadingCoeff_mul, RingHom.map_mul, mul_assoc, mul_assoc]
  

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ C (leadingCoeff p) ∣
    C (leadingCoeff p) * (X ^ natDegree p * q) - C (leadingCoeff p) * (C (leadingCoeff q) * X ^ natDegree (p * q))

Next tactic:
apply dvd_sub (Dvd.intro _ rfl) (Dvd.intro _ rfl)
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  

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

Next tactic:
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]
        apply p.primPart_ne_zero
@[simp]
theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by
  classical
    

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

Next tactic:
suffices h :
      ∀ (n : ℕ) (p q : R[X]), (p * q).degree < n → (p * q).content = p.content * q.content
@[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
    · 

Proof state:
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h : ∀ (n : ℕ) (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
⊢ content (p * q) = content p * content q

Next tactic:
apply h
@[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
      

Proof state:
case a
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h : ∀ (n : ℕ) (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
⊢ degree (p * q) < ↑?n
case n
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
h : ∀ (n : ℕ) (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q
⊢ ℕ

Next tactic:
apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
@[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 _))
    

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ ∀ (n : ℕ) (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q

Next tactic:
intro n
@[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
    

Proof state:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
n : ℕ
⊢ ∀ (p q : R[X]), degree (p * q) < ↑n → content (p * q) = content p * content q

Next tactic:
induction' n with n ih
@[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
    · 

Proof state:
case h.zero
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
⊢ ∀ (p q : R[X]), degree (p * q) < ↑Nat.zero → content (p * q) = content p * content q

Next tactic:
intro p q hpq
@[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
      

Proof state:
case h.zero
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ p q : R[X]
hpq : degree (p * q) < ↑Nat.zero
⊢ content (p * q) = content p * content q

Next tactic:
dsimp at hpq
@[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
      

Proof state:
case h.zero
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ p q : R[X]
hpq : degree (p * q) < ↑0
⊢ content (p * q) = content p * content q

Next tactic:
rw [Nat.cast_zero,
        Nat.WithBot.lt_zero_iff, degree_eq_bot, mul_eq_zero] at hpq
@[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
      

Proof state:
case h.zero
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q✝ p q : R[X]
hpq : p = 0 ∨ q = 0
⊢ content (p * q) = content p * content q

Next tactic:
rcases hpq with (rfl | rfl)
@[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) <;> 

Proof state:
case h.zero.inl
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q✝ q : R[X]
⊢ content (0 * q) = content 0 * content q

Next tactic:
simp
@[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) <;> 

Proof state:
case h.zero.inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p✝ q p : R[X]
⊢ content (p * 0) = content p * content 0

Next tactic:
simp
@[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
    

Proof state:
case h.succ
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]), degree (p * q) < ↑(Nat.succ n) → content (p * q) = content p * content q

Next tactic:
intro p q hpq
@[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
    

Proof state:
case h.succ
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]
hpq : degree (p * q) < ↑(Nat.succ n)
⊢ content (p * q) = content p * content q

Next tactic:
by_cases p0 : p = 0
@[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
    · 

Proof state:
case pos
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]
hpq : degree (p * q) < ↑(Nat.succ n)
p0 : p = 0
⊢ content (p * q) = content p * content q

Next tactic:
simp [p0]
@[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]
    

Proof state:
case neg
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]
hpq : degree (p * q) < ↑(Nat.succ n)
p0 : ¬p = 0
⊢ content (p * q) = content p * content q

Next tactic:
by_cases q0 : q = 0
@[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
    · 

Proof state:
case pos
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]
hpq : degree (p * q) < ↑(Nat.succ n)
p0 : ¬p = 0
q0 : q = 0
⊢ content (p * q) = content p * content q

Next tactic:
simp [q0]
@[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]
    

Proof state:
case neg
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]
hpq : degree (p * q) < ↑(Nat.succ n)
p0 : ¬p = 0
q0 : ¬q = 0
⊢ content (p * q) = content p * content q

Next tactic:
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
@[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
    

Proof state:
case neg
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]
hpq : degree (p * q) < ↑n ∨ natDegree p + natDegree q = n
p0 : ¬p = 0
q0 : ¬q = 0
⊢ content (p * q) = content p * content q

Next tactic:
rcases hpq with (hlt | heq)