awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hdp : d ≤ rootMultiplicity 1 p
hp : ¬p = 0
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
neg :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      d =
    0
neg₁ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      (0 + d) =
    0
neg₂ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d + d)).hilbertPoly
      (0 + d) =
    0
neg₃ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          (HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) * HSub.hSub (α := F[X]) 1 X ^ d)).hilbertPoly
      (0 + d) =
    0
⊢ p.hilbertPoly d = 0
have neg₄ : (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) *
        HSub.hSub (α := F[X]) 1 X ^ d).hilbertPoly
    (0 + d) =
  0
Tactic: rw [heq, ← zero_add d, ← Nat.sub_add_cancel hdp, pow_add (1 - X), ← mul_assoc,
      hilbertPoly_mul_one_sub_pow_add, hilbertPoly]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hdp : d ≤ rootMultiplicity 1 p
hp : ¬p = 0
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
neg :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      d =
    0
neg₁ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      (0 + d) =
    0
neg₂ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d + d)).hilbertPoly
      (0 + d) =
    0
neg₃ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          (HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) * HSub.hSub (α := F[X]) 1 X ^ d)).hilbertPoly
      (0 + d) =
    0
neg₄ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
            HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) *
          HSub.hSub (α := F[X]) 1 X ^ d).hilbertPoly
      (0 + d) =
    0
⊢ p.hilbertPoly d = 0
have neg₅ : (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
        HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d)).hilbertPoly
    0 =
  0
Tactic: rw [heq, ← zero_add d, ← Nat.sub_add_cancel hdp, pow_add (1 - X), ← mul_assoc,
      hilbertPoly_mul_one_sub_pow_add, hilbertPoly]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hdp : d ≤ rootMultiplicity 1 p
hp : ¬p = 0
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
neg :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      d =
    0
neg₁ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p).hilbertPoly
      (0 + d) =
    0
neg₂ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d + d)).hilbertPoly
      (0 + d) =
    0
neg₃ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          (HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) * HSub.hSub (α := F[X]) 1 X ^ d)).hilbertPoly
      (0 + d) =
    0
neg₄ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
            HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d) *
          HSub.hSub (α := F[X]) 1 X ^ d).hilbertPoly
      (0 + d) =
    0
neg₅ :
  (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p - d + d) *
          HSub.hSub (α := F[X]) 1 X ^ (rootMultiplicity 1 p - d)).hilbertPoly
      0 =
    0
⊢ p.hilbertPoly d = 0
have neg₆ : Eq.{u_1 + 1} (α := F[X]) 0 0
Tactic: rw [heq, ← zero_add d, ← Nat.sub_add_cancel hdp, pow_add (1 - X), ← mul_assoc,
      hilbertPoly_mul_one_sub_pow_add, hilbertPoly]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
have rcases : ∀ (q : F[X]) (hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q)
  (hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q),
  (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
Tactic: rcases exists_eq_pow_rootMultiplicity_mul_and_not_dvd p hp 1 with ⟨q, hq1, hq2⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
⊢ p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
have simp : p = q * HSub.hSub (α := F[X]) X 1 ^ rootMultiplicity 1 p
Tactic: simp only [mul_assoc, ← mul_pow, neg_mul, one_mul, neg_sub]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
have heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
Tactic: have heq : p = q * (-1) ^ p.rootMultiplicity 1 * (1 - X) ^ p.rootMultiplicity 1 := by
    simp only [mul_assoc, ← mul_pow, neg_mul, one_mul, neg_sub]
    exact hq1.trans (mul_comm _ _)
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
have nth_rw : ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).hilbertPoly (d - rootMultiplicity 1 p - 1 + 1)).natDegree =
  d - rootMultiplicity 1 p - 1
Tactic: nth_rw 1 [heq, ← Nat.sub_add_cancel (le_of_lt hpd), hilbertPoly_mul_one_sub_pow_add,
    ← Nat.sub_add_cancel (Nat.le_sub_of_add_le' <| add_one_le_of_lt hpd)]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
⊢ natDegree.{u_1} (R := F)
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i)) =
    d - rootMultiplicity 1 p - 1
have pn : natDegree.{u_1} (R := F)
    (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
    | 0 => 0
    | d.succ =>
      ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
        HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
          (preHilbertPoly F d i)) ≤
  d - rootMultiplicity 1 p - 1
Tactic: apply natDegree_eq_of_le_of_coeff_ne_zero
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
pn :
  natDegree.{u_1} (R := F)
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i)) ≤
    d - rootMultiplicity 1 p - 1
⊢ natDegree.{u_1} (R := F)
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i)) =
    d - rootMultiplicity 1 p - 1
have p1 : Ne.{u_1 + 1} (α := F)
  (coeff
    (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
    | 0 => 0
    | d.succ =>
      ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
        HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
          (preHilbertPoly F d i))
    (d - rootMultiplicity 1 p - 1))
  0
Tactic: apply natDegree_eq_of_le_of_coeff_ne_zero
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
⊢ natDegree.{u_1} (R := F)
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i)) ≤
    d - rootMultiplicity 1 p - 1
have apply : ∀ x ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
  (HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff x)
        (preHilbertPoly F (d - rootMultiplicity 1 p - 1) x)).natDegree ≤
    d - rootMultiplicity 1 p - 1
Tactic: apply natDegree_sum_le_of_forall_le _ _ <| fun _ _ => ?_
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
x : ℕ
h : x ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support
⊢ (HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff x)
        (preHilbertPoly F (d - rootMultiplicity 1 p - 1) x)).natDegree ≤
    d - rootMultiplicity 1 p - 1
have apply : natDegree (preHilbertPoly F (d - rootMultiplicity 1 p - 1) x) ≤ d - rootMultiplicity 1 p - 1
Tactic: apply le_trans (natDegree_smul_le _ _)
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
x : ℕ
h : x ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support
⊢ natDegree (preHilbertPoly F (d - rootMultiplicity 1 p - 1) x) ≤ d - rootMultiplicity 1 p - 1
have rw : d - rootMultiplicity 1 p - 1 ≤ d - rootMultiplicity 1 p - 1
Tactic: rw [natDegree_preHilbertPoly]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
⊢ Ne.{u_1 + 1} (α := F)
    (coeff
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i))
      (d - rootMultiplicity 1 p - 1))
    0
have this : Eq.{u_1 + 1} (α := (x : ℕ) → (a : F) → F) (fun (x : ℕ) (a : F) ↦ a) fun (x : ℕ) (a : F) ↦ a * HPow.hPow (α := F) 1 x
Tactic: have : (fun (x : ℕ) (a : F) => a) = fun x a => a * 1 ^ x := by simp only [one_pow, mul_one]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hp : p ≠ 0
hpd : rootMultiplicity 1 p < d
q : F[X]
hq1 : p = HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ^ rootMultiplicity 1 p * q
hq2 : ¬HSub.hSub (α := F[X]) X ((C : (a : F) → F[X]) 1) ∣ q
heq : p = q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p) * HSub.hSub (α := F[X]) 1 X ^ rootMultiplicity 1 p
this :
  Eq.{u_1 + 1} (α := (x : ℕ) → (a : F) → F) (fun (x : ℕ) (a : F) ↦ a) fun (x : ℕ) (a : F) ↦ a * HPow.hPow (α := F) 1 x
⊢ Ne.{u_1 + 1} (α := F)
    (coeff
      (match (motive := ℕ → F[X]) d - rootMultiplicity 1 p - 1 + 1 with
      | 0 => 0
      | d.succ =>
        ∑ i ∈ (q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).support,
          HSMul.hSMul (β := F[X]) ((q * HPow.hPow (α := F[X]) (-1) (rootMultiplicity 1 p)).coeff i)
            (preHilbertPoly F d i))
      (d - rootMultiplicity 1 p - 1))
    0
have p1 : ¬eval 1 q = 0 ∧ ¬(d - rootMultiplicity 1 p - 1)! = 0
Tactic: simp only [finsetSum_coeff, coeff_smul, smul_eq_mul, coeff_preHilbertPoly_self,
      ← Finset.sum_mul, ← sum_def _ (fun _ a => a), this, ← eval_eq_sum, eval_mul, eval_pow,
      eval_neg, eval_one, _root_.mul_eq_zero, pow_eq_zero_iff', neg_eq_zero, one_ne_zero, ne_eq,
      false_and, or_false, inv_eq_zero, cast_eq_zero, not_or]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hh : p.hilbertPoly d ≠ 0
h : p = 0
⊢ False
have hh₁ : Ne.{u_1 + 1} (α := F[X]) (hilbertPoly 0 d) 0
Tactic: rw [h] at hh
Cleared hypotheses: hh, h
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hh : p.hilbertPoly d ≠ 0
⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
have hp : p ≠ 0
Tactic: have hp : p ≠ 0 := by
    intro h
    rw [h] at hh
    exact hh (hilbertPoly_zero_left d)
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hh : p.hilbertPoly d ≠ 0
hp : p ≠ 0
⊢ rootMultiplicity 1 p < d
have by_contra : ∀ (h : ¬rootMultiplicity 1 p < d), False
Tactic: by_contra h
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.HilbertPoly (hypothesis)
Goal:
F : Type u_1
inst : Field F
inst_1 : CharZero F
p : F[X]
d : ℕ
hh : p.hilbertPoly d ≠ 0
hp : p ≠ 0
⊢ (p.hilbertPoly d).natDegree = d - rootMultiplicity 1 p - 1
have hpd : rootMultiplicity 1 p < d
Tactic: have hpd : p.rootMultiplicity 1 < d := by
    by_contra h
    exact hh (hilbertPoly_eq_zero_of_le_rootMultiplicity_one <| not_lt.1 h)
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
⊢ Membership.mem (γ := Ideal R[X][X])
      (Ideal.span
        (insert (α := R[X][X])
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X])
          {HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X])}))
      P ↔
    eval a (eval b P) = 0
have rw : (∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P) ↔
  eval a (eval b P) = 0
Tactic: rw [Ideal.mem_span_pair]
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
⊢ (∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
      HMul.hMul (β := R[X][X]) a_1
            ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) :
              R[X][X]) +
          b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
        P) ↔
    eval a (eval b P) = 0
have mp : (∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P) →
  eval a (eval b P) = 0
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
mp :
  (∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
      HMul.hMul (β := R[X][X]) a_1
            ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) :
              R[X][X]) +
          b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
        P) →
    eval a (eval b P) = 0
⊢ (∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
      HMul.hMul (β := R[X][X]) a_1
            ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) :
              R[X][X]) +
          b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
        P) ↔
    eval a (eval b P) = 0
have mpr : eval a (eval b P) = 0 →
  ∃ (a_2 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_2
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
h :
  ∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
⊢ eval a (eval b P) = 0
have mp : ∀ (w w_1 : R[X][X]),
  eval a
      (eval b
        (HMul.hMul (β := R[X][X]) w
            ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) :
              R[X][X]) +
          w_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]))) =
    0
Tactic: rcases h with ⟨_, _, rfl⟩
Cleared hypotheses: P, h
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
⊢ ∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
have mpr : ∀ (p : R[X]) (hp : eval b P = HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) * p),
  ∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
Tactic: rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
p : R[X]
hp : eval b P = HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) * p
⊢ ∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
have mpr : ∀ (q : R[X][X])
  (hq :
    HSub.hSub (β := R[X][X]) P ((C : (a : R[X]) → R[X][X]) (eval b P) : R[X][X]) =
      HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) * q),
  ∃ (a_1 : R[X][X]) (b_1 : R[X][X]),
    HMul.hMul (β := R[X][X]) a_1
          ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
        b_1 * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
      P
Tactic: rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.Ideal (goal)
Goal:
R : Type u_1
inst : CommRing R
a : R
b : R[X]
P : R[X][X]
h : eval a (eval b P) = 0
p : R[X]
hp : eval b P = HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X]) * p
q : R[X][X]
hq :
  HSub.hSub (β := R[X][X]) P ((C : (a : R[X]) → R[X][X]) (eval b P) : R[X][X]) =
    HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) * q
⊢ HMul.hMul (α := R[X][X]) (β := R[X][X]) ((C : (a : R[X]) → R[X][X]) p : R[X][X])
        ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X]) +
      q * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
    P
have rw : HMul.hMul (α := R[X][X]) (β := R[X][X])
      ((C : (a : R[X]) → R[X][X]) (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) a : R[X])) : R[X][X])
      ((C : (a : R[X]) → R[X][X]) p : R[X][X]) +
    q * HSub.hSub (α := R[X][X]) (β := R[X][X]) X ((C : (a : R[X]) → R[X][X]) b : R[X][X]) =
  P
Tactic: rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]