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 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
⊢ ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have 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
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]
-- 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
⊢ ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have 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
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]
-- 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]
hq : q ≠ 0
⊢ p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart
have mp : p ∣ q → p.content ∣ q.content ∧ p.primPart ∣ q.primPart
Tactic: constructor
-- 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]
hq : q ≠ 0
mp : p ∣ q → p.content ∣ q.content ∧ p.primPart ∣ q.primPart
⊢ p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart
have mpr : p.content ∣ q.content ∧ p.primPart ∣ q.primPart → p ∣ q
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p r : R[X]
hq : p * r ≠ 0
⊢ p.content ∣ (p * r).content ∧ p.primPart ∣ (p * r).primPart
have mp : p.content ∣ p.content * r.content ∧ p.primPart ∣ p * r
Tactic: rw [content_mul, p.isPrimitive_primPart.dvd_primPart_iff_dvd hq]
-- 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]
hq : q ≠ 0
h₁ : p.content ∣ q.content
h₂ : p.primPart ∣ q.primPart
⊢ p ∣ q
have mpr : 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
Tactic: rw [p.eq_C_content_mul_primPart, q.eq_C_content_mul_primPart]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q : R[X]
⊢ ∃ (c : R[X]), ∀ (d : R[X]), p ∣ d ∧ q ∣ d ↔ c ∣ d
have rcases : ∀ (r : R[X]) (rprim : r.IsPrimitive) (hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s),
  ∃ (c : R[X]), ∀ (d : R[X]), p ∣ d ∧ q ∣ d ↔ c ∣ d
Tactic: rcases exists_primitive_lcm_of_isPrimitive p.isPrimitive_primPart
        q.isPrimitive_primPart with
      ⟨r, rprim, hr⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
⊢ ∃ (c : R[X]), ∀ (d : R[X]), p ∣ d ∧ q ∣ d ↔ c ∣ d
have refine : ∀ (s : R[X]),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: refine ⟨C (lcm p.content q.content) * r, fun s => ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have pos : ∀ (hs : s = 0),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: by_cases hs : s = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
pos :
  ∀ (hs : s = 0),
    p ∣ s ∧ q ∣ s ↔
      HMul.hMul (α := R[X])
          ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r ∣
        s
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : ∀ (hs : ¬s = 0),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: by_cases hs : s = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have pos : ∀
  (hpq :
    Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: by_cases hpq : C (lcm p.content q.content) = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
pos :
  ∀
    (hpq :
      Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        (0 : R[X])),
    p ∣ s ∧ q ∣ s ↔
      HMul.hMul (α := R[X])
          ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r ∣
        s
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : ∀
  (hpq :
    ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        (0 : R[X])),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: by_cases hpq : C (lcm p.content q.content) = 0
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
    (0 : R[X])
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have hpq₁ : p = 0 ∨ q = 0
Tactic: rw [C_eq_zero, lcm_eq_zero_iff, content_eq_zero_iff, content_eq_zero_iff] at hpq
Cleared hypotheses: hpq
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq : p = 0 ∨ q = 0
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have pos : ∀ (hpq : p = 0),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: rcases hpq with (hpq | hpq)
Cleared hypotheses: hpq
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq : p = 0 ∨ q = 0
pos :
  ∀ (hpq : p = 0),
    p ∣ s ∧ q ∣ s ↔
      HMul.hMul (α := R[X])
          ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r ∣
        s
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have pos₁ : ∀ (hpq : q = 0),
  p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
Tactic: rcases hpq with (hpq | hpq)
Cleared hypotheses: hpq
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q ∣ s ↔
  HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      r ∣
    s
Tactic: rw [dvd_iff_content_dvd_content_and_primPart_dvd_primPart hs]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
  HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      r ∣
    s
Tactic: rw [dvd_iff_content_dvd_content_and_primPart_dvd_primPart hs]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
  (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
          r).content ∣
      s.content ∧
    (HMul.hMul (α := R[X])
          ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).primPart ∣
      s.primPart
Tactic: rw [dvd_iff_content_dvd_content_and_primPart_dvd_primPart hs]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
⊢ p ∣ s ∧ q ∣ s ↔
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
        r ∣
      s
have neg : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
  (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
          r).content ∣
      s.content ∧
    (HMul.hMul (α := R[X])
          ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).primPart ∣
      s.primPart
Tactic: iterate 3 rw [dvd_iff_content_dvd_content_and_primPart_dvd_primPart hs]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    (HMul.hMul (α := R[X])
            ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).content ∣
        s.content ∧
      (HMul.hMul (α := R[X])
            ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).primPart ∣
        s.primPart
have a : Nontrivial R
Tactic: nontriviality R
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
h : Nontrivial R
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    (HMul.hMul (α := R[X])
            ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).content ∣
        s.content ∧
      (HMul.hMul (α := R[X])
            ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X]) r).primPart ∣
        s.primPart
have rw : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
  (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
Tactic: rw [content_mul, rprim.content_eq_one, mul_one, content_C, normalize_lcm, lcm_dvd_iff,
      primPart_mul (mul_ne_zero hpq rprim.ne_zero), rprim.primPart_eq,
      (isUnit_primPart_C (lcm p.content q.content)).mul_left_dvd, ← hr s.primPart]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
h : Nontrivial R
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
have mp : (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart →
  (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
Tactic: tauto
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
h : Nontrivial R
mp :
  (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart →
    (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
have mpr : (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart →
  (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart
Tactic: tauto
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
this : DecidableEq R := Classical.decEq R
p q r : R[X]
rprim : r.IsPrimitive
hr : ∀ (s : R[X]), p.primPart ∣ s ∧ q.primPart ∣ s ↔ r ∣ s
s : R[X]
hs : ¬s = 0
hpq :
  ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (@lcm _ CommSemiring.toCommMonoidWithZero _ p.content q.content) : R[X])
      (0 : R[X])
h : Nontrivial R
mp :
  (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart →
    (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
mpr :
  (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart →
    (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart
⊢ (p.content ∣ s.content ∧ p.primPart ∣ s.primPart) ∧ q.content ∣ s.content ∧ q.primPart ∣ s.primPart ↔
    (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
have mp₁ : p.content ∣ s.content →
  p.primPart ∣ s.primPart →
    q.content ∣ s.content →
      q.primPart ∣ s.primPart →
        (p.content ∣ s.content ∧ q.content ∣ s.content) ∧ p.primPart ∣ s.primPart ∧ q.primPart ∣ s.primPart
Tactic: tauto