awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsOrderedRing R
P : R[X]
hsc :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
    ((fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
⊢ ∀ (l : List SignType),
    Eq (α := List SignType) (List.map (fun (x : SignType) ↦ -x) (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) l))
      (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) (List.map (fun (x : SignType) ↦ -x) l))

Tactic:
have h_neg_destutter (l : List SignType) :
      (l.destutter (¬· = ·)).map (- ·) = (l.map (- ·)).destutter (¬· = ·) := by
    grind [List.map_destutter, neg_inj]
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsOrderedRing R
P : R[X]
hsc :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
    ((fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
h_neg_destutter :
  ∀ (l : List SignType),
    Eq (α := List SignType) (List.map (fun (x : SignType) ↦ -x) (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) l))
      (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) (List.map (fun (x : SignType) ↦ -x) l))
rw :
  List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2)
          (List.map (⇑SignType.sign : (a : R) → SignType)
            (List.filter
              ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘
                (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
              P.coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.map (⇑SignType.sign : (a : R) → SignType)
            (List.filter ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType))
              P.coeffList))) -
      1
⊢ List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.map (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
            (List.filter
              ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘
                Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
              P.coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.map (⇑SignType.sign : (a : R) → SignType)
            (List.filter ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType))
              P.coeffList))) -
      1

Tactic:
rw [hsc, List.comp_map, ← h_neg_destutter, List.length_map]

Premises used: List.comp_map, List.length_map
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsOrderedRing R
P : R[X]
hsc :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
    ((fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
h_neg_destutter :
  ∀ (l : List SignType),
    Eq (α := List SignType) (List.map (fun (x : SignType) ↦ -x) (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) l))
      (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) (List.map (fun (x : SignType) ↦ -x) l))
e_a :
  Eq.{u_1 + 1} (α := R → Bool)
    ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
    ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType))
⊢ List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2)
          (List.map (⇑SignType.sign : (a : R) → SignType)
            (List.filter
              ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘
                (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
              P.coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.map (⇑SignType.sign : (a : R) → SignType)
            (List.filter ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType))
              P.coeffList))) -
      1

Tactic:
congr 5
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsOrderedRing R
P : R[X]
hsc :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
    ((fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
h_neg_destutter :
  ∀ (l : List SignType),
    Eq (α := List SignType) (List.map (fun (x : SignType) ↦ -x) (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) l))
      (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) (List.map (fun (x : SignType) ↦ -x) l))
e_a :
  ∀ (x : R),
    Eq (α := Bool)
      (((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType)) x)
      (((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType)) x)
⊢ Eq.{u_1 + 1} (α := R → Bool)
    ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
    ((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType))

Tactic:
funext
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsOrderedRing R
P : R[X]
hsc :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ -x)
    ((fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType))
h_neg_destutter :
  ∀ (l : List SignType),
    Eq (α := List SignType) (List.map (fun (x : SignType) ↦ -x) (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) l))
      (List.destutter (fun (x1 x2 : SignType) ↦ ¬x1 = x2) (List.map (fun (x : SignType) ↦ -x) l))
x : R
⊢ Eq (α := Bool)
    (((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (fun (x : SignType) ↦ -x) ∘ (⇑SignType.sign : (a : R) → SignType)) x)
    (((fun (x : SignType) ↦ decide (x ≠ 0)) ∘ (⇑SignType.sign : (a : R) → SignType)) x)

Tactic:
simp [SignType.sign]

Premises used: SignType.sign
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
inr :
  ∀
    (this :
      ∀ {R : Type u_1} [inst : Ring R] [inst_3 : LinearOrder R] [IsStrictOrderedRing R] {η : R} (P : R[X]) (hx : η ≠ 0)
        (hx2 : 0 < η), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations)
    (hx2 : η ≤ 0), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations
wlog! :
  ∀ {R : Type u_1} [inst : Ring R] [inst_3 : LinearOrder R] [IsStrictOrderedRing R] {η : R} (P : R[X]) (hx : η ≠ 0)
    (hx2 : 0 < η), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations
⊢ (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations

Tactic:
wlog! hx2 : 0 < η
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
this :
  ∀ {R : Type u_1} [inst : Ring R] [inst_3 : LinearOrder R] [IsStrictOrderedRing R] {η : R} (P : R[X]) (hx : η ≠ 0)
    (hx2 : 0 < η), (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations
hx2 : η ≤ 0
⊢ (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).signVariations = P.signVariations

Tactic:
simpa [lt_of_le_of_ne hx2, hx] using this (η := -η) (P := -P)

Premises used: Neg.neg, lt_of_le_of_ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R_1 : Type u_1
inst : Ring R_1
inst_1 : LinearOrder R_1
η_1 : R_1
R : Type u_1
inst_2 : Ring R
inst_3 : LinearOrder R
inst_4 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
hx2 : 0 < η
rw :
  List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ η * x)
              P.coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (⇑SignType.sign : (a : R) → SignType) P.coeffList))) -
      1
⊢ List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (⇑SignType.sign : (a : R) → SignType)
              (HMul.hMul (α := R[X]) ((C : (a : R) → R[X]) η : R[X]) P).coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (⇑SignType.sign : (a : R) → SignType) P.coeffList))) -
      1

Tactic:
rw [coeffList_C_mul _ (lt_or_lt_iff_ne.mp (.inr hx2)), ← List.comp_map]

Premises used: Polynomial.coeffList_C_mul, List.comp_map, Iff.mp, lt_or_lt_iff_ne, Or.inr
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R_1 : Type u_1
inst : Ring R_1
inst_1 : LinearOrder R_1
η_1 : R_1
R : Type u_1
inst_2 : Ring R
inst_3 : LinearOrder R
inst_4 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
hx2 : 0 < η
e_a :
  Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ η * x)
    (⇑SignType.sign : (a : R) → SignType)
⊢ List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ η * x)
              P.coeffList))) -
      1 =
    List.length (α := SignType)
        (List.destutter (fun (x1 x2 : SignType) ↦ x1 ≠ x2)
          (List.filter (fun (x : SignType) ↦ decide (x ≠ 0))
            (List.map (⇑SignType.sign : (a : R) → SignType) P.coeffList))) -
      1

Tactic:
congr 5
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R_1 : Type u_1
inst : Ring R_1
inst_1 : LinearOrder R_1
η_1 : R_1
R : Type u_1
inst_2 : Ring R
inst_3 : LinearOrder R
inst_4 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
hx2 : 0 < η
e_a :
  ∀ (x : R),
    Eq (α := SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) (fun (x : R) ↦ η * x) x : SignType)
      ((SignType.sign : (a : R) → SignType) x : SignType)
⊢ Eq.{u_1 + 1} (α := R → SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) fun (x : R) ↦ η * x)
    (⇑SignType.sign : (a : R) → SignType)

Tactic:
funext
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R_1 : Type u_1
inst : Ring R_1
inst_1 : LinearOrder R_1
η_1 : R_1
R : Type u_1
inst_2 : Ring R
inst_3 : LinearOrder R
inst_4 : IsStrictOrderedRing R
η : R
P : R[X]
hx : η ≠ 0
hx2 : 0 < η
x : R
⊢ Eq (α := SignType) (Function.comp (β := R) (⇑SignType.sign : (a : R) → SignType) (fun (x : R) ↦ η * x) x : SignType)
    ((SignType.sign : (a : R) → SignType) x : SignType)

Tactic:
simp [hx2, sign_mul]

Premises used: sign_mul
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
obtain :
  ∀ (d : ℕ) (hd : P.natDegree = d + 1),
    (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).eraseLead.signVariations =
      (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).signVariations
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).eraseLead.signVariations =
    (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).signVariations

Tactic:
obtain ⟨d, hd⟩ := Nat.exists_eq_add_one.mpr (natDegree_pos_of_nextCoeff_ne_zero hc.ne)

Premises used: Iff.mpr, Polynomial.natDegree_pos_of_nextCoeff_ne_zero, Nat.exists_eq_add_one, LT.lt.ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
⊢ P ≠ 0

Tactic:
have hPn0 : P ≠ 0 :=
      leadingCoeff_ne_zero.mp hP₀.ne'
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hPn0 : P ≠ 0
rw : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X])).natDegree + P.natDegree = P.natDegree + 1
rw₁ : 1 + P.natDegree = P.natDegree + 1
rw₂ rw₃ : P.natDegree + 1 = P.natDegree + 1
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1

Tactic:
rw [natDegree_mul (X_sub_C_ne_zero η) hPn0, natDegree_X_sub_C, add_comm]

Premises used: Polynomial.natDegree_mul, Polynomial.natDegree_X_sub_C, add_comm, Polynomial.X_sub_C_ne_zero
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1

Tactic:
have hndxP : natDegree ((X - C η) * P) = P.natDegree + 1 := by
    have hPn0 : P ≠ 0 :=
      leadingCoeff_ne_zero.mp hP₀.ne'
    rw [natDegree_mul (X_sub_C_ne_zero η) hPn0, natDegree_X_sub_C, add_comm]
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
⊢ P.eraseLead ≠ 0

Tactic:
have hePn0 : P.eraseLead ≠ 0 :=
      mt nextCoeff_eq_zero_of_eraseLead_eq_zero hc.ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hePn0 : P.eraseLead ≠ 0
rw : P.eraseLead.natDegree + 1 = P.natDegree
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree

Tactic:
rw [natDegree_mul (X_sub_C_ne_zero η) hePn0, natDegree_X_sub_C, add_comm]

Premises used: Polynomial.natDegree_mul, Polynomial.natDegree_X_sub_C, add_comm, Polynomial.X_sub_C_ne_zero
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hePn0 : P.eraseLead ≠ 0
⊢ P.eraseLead.natDegree + 1 = P.natDegree

Tactic:
natDegree_eraseLead_add_one hc.ne

Premises used: Polynomial.natDegree_eraseLead_add_one, LT.lt.ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hePn0 : P.eraseLead ≠ 0
⊢ P.eraseLead.natDegree + 1 = P.natDegree

Tactic:
exact natDegree_eraseLead_add_one hc.ne

Premises used: Polynomial.natDegree_eraseLead_add_one, LT.lt.ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree

Tactic:
have hndxeP : natDegree ((X - C η) * P.eraseLead) = P.natDegree := by
    have hePn0 : P.eraseLead ≠ 0 :=
      mt nextCoeff_eq_zero_of_eraseLead_eq_zero hc.ne
    rw [natDegree_mul (X_sub_C_ne_zero η) hePn0, natDegree_X_sub_C, add_comm]
    exact natDegree_eraseLead_add_one hc.ne
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)

Tactic:
grind [nextCoeff_of_natDegree_pos, coeff_X_sub_C_mul]

Premises used: Polynomial.nextCoeff_of_natDegree_pos, Polynomial.coeff_X_sub_C_mul
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)

Tactic:
have hQ : ((X - C η) * P).nextCoeff = coeff P d - η * coeff P (d + 1) := by
    grind [nextCoeff_of_natDegree_pos, coeff_X_sub_C_mul]
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
rw : P.coeff d < η * P.coeff (d + 1)
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff < 0

Tactic:
rw [hQ, sub_neg]

Premises used: sub_neg
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
trans : P.coeff d < 0
trans₁ : 0 < η * P.coeff (d + 1)
⊢ P.coeff d < η * P.coeff (d + 1)

Tactic:
trans 0
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ P.coeff d < 0

Tactic:
grind [nextCoeff_of_natDegree_pos]

Premises used: Polynomial.nextCoeff_of_natDegree_pos
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ 0 < η * P.coeff (d + 1)

Tactic:
hd ▸ mul_pos hη hP₀

Premises used: Eq.rec, mul_pos
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ 0 < η * P.coeff (d + 1)

Tactic:
exact hd ▸ mul_pos hη hP₀
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff < 0

Tactic:
have hQ₁ : ((X - C η) * P).nextCoeff < 0 := by
    rw [hQ, sub_neg]
    trans 0
    · grind [nextCoeff_of_natDegree_pos]
    · exact hd ▸ mul_pos hη hP₀
-- Module: Mathlib.Algebra.Polynomial.RuleOfSigns
Goal:
R : Type u_1
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
P : R[X]
η : R
hη : 0 < η
hP₀ : 0 < P.leadingCoeff
hc : P.nextCoeff < 0
d : ℕ
hd : P.natDegree = d + 1
hndxP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).natDegree = P.natDegree + 1
hndxeP : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P.eraseLead).natDegree = P.natDegree
hQ :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff = P.coeff d - η * P.coeff (d + 1)
hQ₁ : (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).nextCoeff < 0
apply :
  (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).eraseLead.natDegree + 1 = P.natDegree + 1
⊢ (HSub.hSub (α := R[X]) (β := R[X]) X ((C : (a : R) → R[X]) η : R[X]) * P).eraseLead.natDegree = P.natDegree

Tactic:
apply Nat.add_right_cancel (m := 1)

Premises used: Nat.add_right_cancel