l3lab/ntp-mathlib huggingface dataset:
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
    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]
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ cancelLeads r s
⊢ r ∣ primPart s

Next tactic:
rw [cancelLeads, tsub_eq_zero_iff_le.mpr hs, pow_zero, mul_one] at rcs
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
    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
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
⊢ r ∣ primPart s

Next tactic:
have h :=
      dvd_add rcs (Dvd.intro_left (C (leadingCoeff s) * X ^ (natDegree s - natDegree r)) 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
    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)
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h :
  r ∣
    C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +
      C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
⊢ r ∣ primPart s

Next tactic:
have hC0 := rprim.ne_zero
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
    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
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h :
  r ∣
    C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +
      C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
hC0 : r ≠ 0
⊢ r ∣ primPart s

Next tactic:
rw [Ne.def, ← leadingCoeff_eq_zero, ← C_eq_zero] at hC0
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
    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
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h :
  r ∣
    C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r +
      C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
hC0 : ¬C (leadingCoeff r) = 0
⊢ r ∣ primPart s

Next tactic:
rw [sub_add_cancel, ← rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)] at h
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
    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
    

Proof state:
case neg
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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h : r ∣ primPart (C (leadingCoeff r) * s)
hC0 : ¬C (leadingCoeff r) = 0
⊢ r ∣ primPart s

Next tactic:
rcases isUnit_primPart_C r.leadingCoeff with ⟨u, hu⟩
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
    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⟩
    

Proof state:
case neg.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
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h : r ∣ primPart (C (leadingCoeff r) * s)
hC0 : ¬C (leadingCoeff r) = 0
u : R[X]ˣ
hu : ↑u = primPart (C (leadingCoeff r))
⊢ r ∣ primPart s

Next tactic:
apply h.trans (Associated.symm ⟨u, _⟩).dvd
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
    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
    

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 : R[X]
rdeg : natDegree r = Nat.find h✝
rprim : IsPrimitive r
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, natDegree s = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : natDegree s = Nat.find con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : ¬natDegree s ≤ 0
hcancel : natDegree (cancelLeads r s) < Nat.find con
rcs : r ∣ C (leadingCoeff r) * s - C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r
h : r ∣ primPart (C (leadingCoeff r) * s)
hC0 : ¬C (leadingCoeff r) = 0
u : R[X]ˣ
hu : ↑u = primPart (C (leadingCoeff r))
⊢ primPart s * ↑u = primPart (C (leadingCoeff r) * s)

Next tactic:
rw [primPart_mul (mul_ne_zero hC0 s0), hu, mul_comm]
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ 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]
hq : q ≠ 0
⊢ p ∣ q ↔ content p ∣ content q ∧ primPart p ∣ primPart q

Next tactic:
constructor
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> 

Proof state:
case mp
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hq : q ≠ 0
⊢ p ∣ q → content p ∣ content q ∧ primPart p ∣ primPart q

Next tactic:
intro h
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> 

Proof state:
case mpr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hq : q ≠ 0
⊢ content p ∣ content q ∧ primPart p ∣ primPart q → p ∣ q

Next tactic:
intro h
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> intro h
  · 

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

Next tactic:
rcases h with ⟨r, rfl⟩
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> intro h
  · rcases h with ⟨r, rfl⟩
    

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

Next tactic:
rw [content_mul, p.isPrimitive_primPart.dvd_primPart_iff_dvd hq]
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> intro h
  · rcases h with ⟨r, rfl⟩
    rw [content_mul, p.isPrimitive_primPart.dvd_primPart_iff_dvd hq]
    

Proof state:
case mp.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p r : R[X]
hq : p * r ≠ 0
⊢ content p ∣ content p * content r ∧ primPart p ∣ p * r

Next tactic:
exact ⟨Dvd.intro _ rfl, p.primPart_dvd.trans (Dvd.intro _ rfl)⟩
theorem dvd_iff_content_dvd_content_and_primPart_dvd_primPart {p q : R[X]} (hq : q ≠ 0) :
    p ∣ q ↔ p.content ∣ q.content ∧ p.primPart ∣ q.primPart := by
  constructor <;> intro h
  · rcases h with ⟨r, rfl⟩
    rw [content_mul, p.isPrimitive_primPart.dvd_primPart_iff_dvd hq]
    exact ⟨Dvd.intro _ rfl, p.primPart_dvd.trans (Dvd.intro _ rfl)⟩
  · 

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

Next tactic:
rw [p.eq_C_content_mul_primPart, q.eq_C_content_mul_primPart]