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
    

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

Next tactic:
rcases Nat.find_spec con with ⟨s, sdeg, ⟨ps, qs⟩, rs⟩
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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⟩
    

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

Next tactic:
have s0 : s ≠ 0 := by
      contrapose! rs
      simp [rs]
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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
      

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
⊢ s ≠ 0

Next tactic:
contrapose! rs
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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
      

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
ps : p ∣ s
qs : q ∣ s
rs : s = 0
⊢ r ∣ s

Next tactic:
simp [rs]
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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]
    

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

Next tactic:
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⟩
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⟩
    

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

Next tactic:
rw [← rdeg] at hs
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
    

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

Next tactic:
by_cases sC : s.natDegree ≤ 0
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
    · 

Proof state:
case pos
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
⊢ False

Next tactic:
rw [eq_C_of_natDegree_le_zero (le_trans hs sC), isPrimitive_iff_content_eq_one, content_C,
        normalize_eq_one] at rprim
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
      

Proof state:
case pos
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 : IsUnit (coeff r 0)
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
⊢ False

Next tactic:
rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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
      

Proof state:
case pos
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 : IsUnit (coeff r 0)
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 : ¬coeff r 0 ∣ content s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : natDegree r ≤ natDegree s
sC : natDegree s ≤ 0
⊢ False

Next tactic:
apply rs rprim.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
    

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
⊢ False

Next tactic:
have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs (lt_of_not_ge sC)
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)
    

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) < natDegree s
⊢ False

Next tactic:
rw [sdeg] at hcancel
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
    

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
⊢ False

Next tactic:
apply Nat.find_min con hcancel
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
    

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
⊢ ∃ s_1, natDegree s_1 = natDegree (cancelLeads r s) ∧ (p ∣ s_1 ∧ q ∣ s_1) ∧ ¬r ∣ s_1

Next tactic:
refine'
      ⟨_, rfl, ⟨dvd_cancelLeads_of_dvd_of_dvd pr ps, dvd_cancelLeads_of_dvd_of_dvd qr qs⟩,
        fun rcs => rs _⟩
theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
    ∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s := by
  classical
    have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r :=
      ⟨(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _⟩
    rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
    refine' ⟨r, rprim, fun s => ⟨_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
    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 _⟩
    

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 ∣ s

Next tactic:
rw [← rprim.dvd_primPart_iff_dvd s0]