awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h_1 : Nontrivial R
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)
h : (q.primPart * p.primPart).content = 1
rw :
  (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
rw₁ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (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
⊢ (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
have rw₂ : HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
    (p.primPart * HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content =
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h_1 : Nontrivial R
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)
h : (q.primPart * p.primPart).content = 1
rw :
  (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
rw₁ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (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
rw₂ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (p.primPart * HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (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 *
        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
have rw₃ : HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart * p.primPart).content =
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h_1 : Nontrivial R
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)
h : (q.primPart * p.primPart).content = 1
rw :
  (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
rw₁ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (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
rw₂ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (p.primPart * HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
rw₃ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart * p.primPart).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (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 *
        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
have rw₄ : HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) (q.primPart * p.primPart)).content =
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h_1 : Nontrivial R
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)
h : (q.primPart * p.primPart).content = 1
rw :
  (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
rw₁ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (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
rw₂ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (p.primPart * HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
rw₃ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart * p.primPart).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
rw₄ :
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
      (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) (q.primPart * p.primPart)).content =
    HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
      (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 *
        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
have rw₅ : HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R)
    (HMul.hMul (α := R) ((normalize : (a : R) → R) q.content : R) (q.primPart * p.primPart).content) =
  HMul.hMul (α := R) ((normalize : (a : R) → R) p.content : R) p.primPart.content *
    (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) q.content : R[X]) q.primPart).content
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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
have neg : (q.primPart * p.primPart).content = 1
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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
have neg : @gcd _ CommSemiring.toCommMonoidWithZero _ (q.primPart * p.primPart).eraseLead.content
    (q.primPart.leadingCoeff * p.primPart.leadingCoeff) ∣
  1
Tactic: rw [← normalize_content, normalize_eq_one, isUnit_iff_dvd_one,
      content_eq_gcd_leadingCoeff_content_eraseLead, leadingCoeff_mul, gcd_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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
have neg : @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
Tactic: apply (gcd_mul_dvd_mul_gcd _ _ _).trans
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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
have neg : (p.primPart.eraseLead * q.primPart).degree < (↑n : WithBot ℕ)
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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)
neg : (p.primPart.eraseLead * 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
have neg₁ : (q.primPart.eraseLead * p.primPart).degree < (↑n : WithBot ℕ)
Tactic: 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]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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)
have neg : p.primPart.eraseLead.degree < p.primPart.degree
Tactic: rw [← heq, degree_mul, WithBot.add_lt_add_iff_right]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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)
neg : p.primPart.eraseLead.degree < p.primPart.degree
⊢ (p.primPart.eraseLead * q.primPart).degree < (↑n : WithBot ℕ)
have neg₁ : q.primPart.degree ≠ ⊥
Tactic: rw [← heq, degree_mul, WithBot.add_lt_add_iff_right]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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.degree ≠ ⊥
have neg : ¬q.primPart = 0
Tactic: rw [Ne, degree_eq_bot]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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)
have neg : q.primPart.eraseLead.degree < q.primPart.degree
Tactic: rw [mul_comm, ← heq, degree_mul, WithBot.add_lt_add_iff_left]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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)
neg : q.primPart.eraseLead.degree < q.primPart.degree
⊢ (q.primPart.eraseLead * p.primPart).degree < (↑n : WithBot ℕ)
have neg₁ : p.primPart.degree ≠ ⊥
Tactic: rw [mul_comm, ← heq, degree_mul, WithBot.add_lt_add_iff_left]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p_1 q_1 : R[X]
h : Nontrivial R
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.degree ≠ ⊥
have neg : ¬p.primPart = 0
Tactic: rw [Ne, degree_eq_bot]
-- 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]
h : Nontrivial R
⊢ (p * q).content = p.content * q.content
have suffices : ∀ (n : ℕ) (p q : R[X]), (p * q).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
-- 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]
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
⊢ (p * q).content = p.content * q.content
have 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
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