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]
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
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
⊢ (p * q).content = p.content * q.content
have pos : ∀ (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)) (p0 : p = 0), (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]
hp : p.IsPrimitive
hq : q.IsPrimitive
⊢ (p * q).IsPrimitive
have rw : (p * q).content = 1
Tactic: rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_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 q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
rw : (p * q).content = 1
⊢ (p * q).IsPrimitive
have rw₁ : p.content * q.content = 1
Tactic: rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_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 q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
rw : (p * q).content = 1
rw₁ : p.content * q.content = 1
⊢ (p * q).IsPrimitive
have rw₂ : 1 * q.content = 1
Tactic: rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_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 q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
rw : (p * q).content = 1
rw₁ : p.content * q.content = 1
rw₂ : 1 * q.content = 1
⊢ (p * q).IsPrimitive
have rw₃ : HMul.hMul (α := R) 1 1 = 1
Tactic: rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_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 q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
rw : (p * q).content = 1
rw₁ : p.content * q.content = 1
rw₂ : 1 * q.content = 1
rw₃ : HMul.hMul (α := R) 1 1 = 1
⊢ (p * q).IsPrimitive
have rw₄ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [isPrimitive_iff_content_eq_one, content_mul, hp.content_eq_one, hq.content_eq_one, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
h0 : p * q ≠ 0
⊢ (p * q).primPart = p.primPart * q.primPart
have h0₁ : ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (0 : R[X])
Tactic: rw [Ne, ← content_eq_zero_iff, ← C_eq_zero] at h0
Cleared hypotheses: h0
-- 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]
h0 : ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (0 : R[X])
⊢ (p * q).primPart = p.primPart * q.primPart
have apply : HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p * q).primPart =
  HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p.primPart * q.primPart)
Tactic: apply mul_left_cancel₀ h0
-- 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]
h0 : ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (0 : R[X])
| HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p * q).primPart
have 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
Tactic: rw [← (p * q).eq_C_content_mul_primPart, 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
p q : R[X]
h0 : ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (0 : R[X])
⊢ HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p * q).primPart =
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p.primPart * q.primPart)
have conv_lhs : 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 =
  HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p.primPart * q.primPart)
Tactic: conv_lhs =>
    rw [← (p * q).eq_C_content_mul_primPart, 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
p q : R[X]
h0 : ¬Eq (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (0 : R[X])
⊢ 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 =
    HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) (p * q).content : R[X]) (p.primPart * q.primPart)
have 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 =
  HMul.hMul (α := R[X]) (β := R[X]) ((C : (a : R) → R[X]) p.content : R[X]) ((C : (a : R) → R[X]) q.content : R[X]) *
    (p.primPart * q.primPart)
Tactic: rw [content_mul, map_mul]
-- 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 ≠ 0
⊢ p ∣ q.primPart ↔ p ∣ q
have refine : ∀ (h : p ∣ q), p ∣ q.primPart
Tactic: refine ⟨fun h => h.trans (Dvd.intro_left _ q.eq_C_content_mul_primPart.symm), fun h => ?_⟩
-- 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 ≠ 0
h : p ∣ q
⊢ p ∣ q.primPart
have rcases : ∀ (r : R[X]) (hq : p * r ≠ 0), p ∣ (p * r).primPart
Tactic: rcases h with ⟨r, rfl⟩
Cleared hypotheses: q, hq, h
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
r : R[X]
hq : p * r ≠ 0
⊢ p ∣ (p * r).primPart
have apply : p * r.primPart = (p * r).primPart
Tactic: apply Dvd.intro _
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
r : R[X]
hq : p * r ≠ 0
⊢ p * r.primPart = (p * r).primPart
have rw : p * r.primPart = p.primPart * r.primPart
Tactic: rw [primPart_mul hq, hp.primPart_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p : R[X]
hp : p.IsPrimitive
r : R[X]
hq : p * r ≠ 0
rw : p * r.primPart = p.primPart * r.primPart
⊢ p * r.primPart = (p * r).primPart
have rw₁ : p * r.primPart = p * r.primPart
Tactic: rw [primPart_mul hq, hp.primPart_eq]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
⊢ ∃ (r : R[X]), r.IsPrimitive ∧ ∀ (s : R[X]), p ∣ s ∧ q ∣ s ↔ r ∣ s
have h : ∃ (n : ℕ) (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
Tactic: 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 _ _⟩
-- 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
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
have rcases : ∀ (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: rcases Nat.find_spec h with ⟨r, rdeg, rprim, pr, qr⟩
-- 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
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
have refine : ∀ (s : R[X]), p ∣ s ∧ q ∣ s → r ∣ s
Tactic: refine ⟨r, rprim, fun s => ⟨?_, fun rs => ⟨pr.trans rs, qr.trans rs⟩⟩⟩
-- 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
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
have suffices : ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s
Tactic: suffices hs : ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s from
      hs s.natDegree s rfl
-- 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
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
⊢ ∀ (n : ℕ) (s : R[X]), s.natDegree = n → p ∣ s ∧ q ∣ s → r ∣ s
have by_contra! : ∀ (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s), False
Tactic: by_contra! con
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
⊢ False
have rcases : ∀ (s : R[X])
  (sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con)
  (rs : ¬r ∣ s) (ps : p ∣ s) (qs : q ∣ s), False
Tactic: rcases Nat.find_spec con with ⟨s, sdeg, ⟨ps, qs⟩, rs⟩
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
⊢ s ≠ 0
have contrapose : ∀ (rs : s = 0), r ∣ s
Tactic: contrapose rs
Cleared hypotheses: rs
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
⊢ False
have s0 : s ≠ 0
Tactic: have s0 : s ≠ 0 := by
      contrapose rs
      simp [rs]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
⊢ False
have hs : Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h ≤ s.natDegree
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⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : Nat.find (p := fun (n : ℕ) ↦ ∃ (r : R[X]), r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r) h ≤ s.natDegree
⊢ False
have hs₁ : r.natDegree ≤ s.natDegree
Tactic: rw [← rdeg] at hs
Cleared hypotheses: hs
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
⊢ False
have pos : ∀ (sC : s.natDegree ≤ 0), False
Tactic: by_cases! sC : s.natDegree ≤ 0
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
pos : ∀ (sC : s.natDegree ≤ 0), False
⊢ False
have neg : ∀ (sC : 0 < s.natDegree), False
Tactic: by_cases! sC : s.natDegree ≤ 0
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : s.natDegree ≤ 0
⊢ False
have pos : ∀ (rprim : IsUnit (r.coeff 0)) (pr : p ∣ r) (qr : q ∣ r)
  (con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) (s : R[X])
  (sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con)
  (rs : ¬r ∣ s) (ps : p ∣ s) (qs : q ∣ s) (s0 : s ≠ 0) (hs : r.natDegree ≤ s.natDegree) (sC : s.natDegree ≤ 0), False
Tactic: rw [eq_C_of_natDegree_le_zero (le_trans hs sC), isPrimitive_iff_content_eq_one, content_C,
        normalize_eq_one] at rprim
Cleared hypotheses: rprim, pr, qr, con, s, sdeg, rs, ps, qs, s0, hs, sC
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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 : IsUnit (r.coeff 0)
pr : p ∣ r
qr : q ∣ r
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : s.natDegree ≤ 0
⊢ False
have rs₁ : ¬r.coeff 0 ∣ s.content
Tactic: rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs
Cleared hypotheses: rs, ps, qs, s0, hs, sC
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : 0 < s.natDegree
⊢ False
have hcancel : (r.cancelLeads s).natDegree < s.natDegree
Tactic: have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs sC
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Content (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : 0 < s.natDegree
hcancel : (r.cancelLeads s).natDegree < s.natDegree
⊢ False
have hcancel₁ : (r.cancelLeads s).natDegree < Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
Tactic: rw [sdeg] at hcancel
Cleared hypotheses: hcancel
-- 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
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
con : ∃ (n : ℕ) (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s
s : R[X]
sdeg : s.natDegree = Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
rs : ¬r ∣ s
ps : p ∣ s
qs : q ∣ s
s0 : s ≠ 0
hs : r.natDegree ≤ s.natDegree
sC : 0 < s.natDegree
hcancel :
  (r.cancelLeads s).natDegree <
    Nat.find (p := fun (n : ℕ) ↦ ∃ (s : R[X]), s.natDegree = n ∧ (p ∣ s ∧ q ∣ s) ∧ ¬r ∣ s) con
⊢ False
have neg : ∃ (s_1 : R[X]), s_1.natDegree = (r.cancelLeads s).natDegree ∧ (p ∣ s_1 ∧ q ∣ s_1) ∧ ¬r ∣ s_1
Tactic: apply Nat.find_min con hcancel