awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u
inst : CommRing R
inst_1 : IsLocalRing R
tfae_3_to_2 :
  ∀
    (a :
      ∀ {K : Type u} [inst_2 : Field K] (φ : R →+* K),
        Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
          ∀ (f : R[X]),
            f.Monic →
              ∀ (a₀ : K),
                eval₂ φ a₀ f = 0 →
                  eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 →
                    ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀)
    (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R),
        Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R) (0 : ResidueField R) →
          Ne (α := ResidueField R)
              ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
              (0 : ResidueField R) →
            ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀
tfae_2_to_1 :
  ∀
    (a :
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : ResidueField R),
            Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R)
                (0 : ResidueField R) →
              Ne (α := ResidueField R)
                  ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
                  (0 : ResidueField R) →
                ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀),
    HenselianLocalRing R
hR : HenselianLocalRing R
K : Type u
_K : Field K
φ : R →+* K
hφ : Surjective (α := R) (β := K) (⇑φ : (a : R) → K)
f : R[X]
hf : f.Monic
a₀ : R
H :
  Eq (α := K) ((φ : (a : R) → K) (Polynomial.eval a₀ f) : K) (0 : K) →
    IsUnit (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) →
      ∃ (a : R), f.IsRoot a ∧ Eq (α := K) ((φ : (a : R) → K) (a - a₀) : K) (0 : K)
h₁ : Eq (α := K) ((φ : (a : R) → K) (Polynomial.eval a₀ f) : K) (0 : K)
h₂ : Ne (α := K) ((φ : (a : R) → K) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : K) (0 : K)
a : R
ha₁ : f.IsRoot a
ha₂ : Eq (α := K) ((φ : (a : R) → K) (a - a₀) : K) (0 : K)
rwa :
  ∀ (ha₂ : HSub.hSub (α := K) (β := K) ((φ : (a : R) → K) a : K) ((φ : (a : R) → K) a₀ : K) = 0),
    Eq (α := K) ((φ : (a : R) → K) a : K) ((φ : (a : R) → K) a₀ : K)
rwa₁ rwa₂ rwa₃ rwa₄ :
  ∀ (ha₂ : Eq (α := K) ((φ : (a : R) → K) a : K) ((φ : (a : R) → K) a₀ : K)),
    Eq (α := K) ((φ : (a : R) → K) a : K) ((φ : (a : R) → K) a₀ : K)
⊢ Eq (α := K) ((φ : (a : R) → K) a : K) ((φ : (a : R) → K) a₀ : K)

Tactic:
rwa [φ.map_sub, sub_eq_zero] at ha₂

Premises used: sub_eq_zero, RingHom.map_sub
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u
inst : CommRing R
inst_1 : IsLocalRing R
tfae_3_to_2 :
  ∀
    (a :
      ∀ {K : Type u} [inst_2 : Field K] (φ : R →+* K),
        Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
          ∀ (f : R[X]),
            f.Monic →
              ∀ (a₀ : K),
                eval₂ φ a₀ f = 0 →
                  eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 →
                    ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀)
    (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R),
        Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R) (0 : ResidueField R) →
          Ne (α := ResidueField R)
              ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
              (0 : ResidueField R) →
            ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀
tfae_2_to_1 :
  ∀
    (a :
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : ResidueField R),
            Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R)
                (0 : ResidueField R) →
              Ne (α := ResidueField R)
                  ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
                  (0 : ResidueField R) →
                ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀),
    HenselianLocalRing R
⊢ ∀ (a : HenselianLocalRing R) {K : Type u} [inst_2 : Field K] (φ : R →+* K),
    Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : K),
            eval₂ φ a₀ f = 0 →
              eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 → ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀

Tactic:
tfae_have 1 → 3
  | hR, K, _K, φ, hφ, f, hf, a₀, h₁, h₂ => by
    obtain ⟨a₀, rfl⟩ := hφ a₀
    have H := HenselianLocalRing.is_henselian f hf a₀
    simp only [← ker_eq_maximalIdeal φ hφ, eval₂_at_apply, RingHom.mem_ker] at H h₁ h₂
    obtain ⟨a, ha₁, ha₂⟩ := H h₁ (by
      contrapose h₂
      rwa [← mem_nonunits_iff, ← mem_maximalIdeal, ← ker_eq_maximalIdeal φ hφ,
        RingHom.mem_ker] at h₂)
    refine ⟨a, ha₁, ?_⟩
    rwa [φ.map_sub, sub_eq_zero] at ha₂
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u
inst : CommRing R
inst_1 : IsLocalRing R
tfae_3_to_2 :
  ∀
    (a :
      ∀ {K : Type u} [inst_2 : Field K] (φ : R →+* K),
        Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
          ∀ (f : R[X]),
            f.Monic →
              ∀ (a₀ : K),
                eval₂ φ a₀ f = 0 →
                  eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 →
                    ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀)
    (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R),
        Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R) (0 : ResidueField R) →
          Ne (α := ResidueField R)
              ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
              (0 : ResidueField R) →
            ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀
tfae_2_to_1 :
  ∀
    (a :
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : ResidueField R),
            Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R)
                (0 : ResidueField R) →
              Ne (α := ResidueField R)
                  ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
                  (0 : ResidueField R) →
                ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀),
    HenselianLocalRing R
tfae_1_to_3 :
  ∀ (a : HenselianLocalRing R) {K : Type u} [inst_2 : Field K] (φ : R →+* K),
    Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : K),
            eval₂ φ a₀ f = 0 →
              eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 → ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀
⊢ [HenselianLocalRing R,
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : ResidueField R),
            Eq (α := ResidueField R) ((aeval a₀ : (a : R[X]) → ResidueField R) f : ResidueField R)
                (0 : ResidueField R) →
              Ne (α := ResidueField R)
                  ((aeval a₀ : (a : R[X]) → ResidueField R) ((derivative : (a : R[X]) → R[X]) f) : ResidueField R)
                  (0 : ResidueField R) →
                ∃ (a : R), f.IsRoot a ∧ (residue R : (a : R) → ResidueField R) a = a₀,
      ∀ {K : Type u} [inst_2 : Field K] (φ : R →+* K),
        Surjective (α := R) (β := K) (⇑φ : (a : R) → K) →
          ∀ (f : R[X]),
            f.Monic →
              ∀ (a₀ : K),
                eval₂ φ a₀ f = 0 →
                  eval₂ φ a₀ ((derivative : (a : R[X]) → R[X]) f) ≠ 0 →
                    ∃ (a : R), f.IsRoot a ∧ (φ : (a : R) → K) a = a₀].TFAE

Tactic:
tfae_finish
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
rw : ∀ (b : Ideal R), Membership.mem (γ := Set (Ideal R)) {J : Ideal R | ⊥ ≤ J ∧ J.IsMaximal} b → maximalIdeal R ≤ b
⊢ LE.le (α := Ideal R) (maximalIdeal R) (Ideal.jacobson ⊥)

Tactic:
rw [Ideal.jacobson, le_sInf_iff]

Premises used: le_sInf_iff, Ideal.jacobson
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
I : Ideal R
hI : I.IsMaximal
⊢ maximalIdeal R ≤ I

Tactic:
(eq_maximalIdeal hI).ge

Premises used: Eq.ge, IsLocalRing.eq_maximalIdeal
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
I : Ideal R
hI : I.IsMaximal
⊢ maximalIdeal R ≤ I

Tactic:
exact (eq_maximalIdeal hI).ge

Premises used: Eq.ge, IsLocalRing.eq_maximalIdeal
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  IsUnit (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
⊢ ∃ (a : R), f.IsRoot a ∧ Membership.mem (γ := Ideal R) (maximalIdeal R) (a - a₀)

Tactic:
HenselianLocalRing.is_henselian f hf a₀ h₁ ?_

Premises used: HenselianLocalRing.is_henselian
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  IsUnit (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
refine : IsUnit (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f))
⊢ ∃ (a : R), f.IsRoot a ∧ Membership.mem (γ := Ideal R) (maximalIdeal R) (a - a₀)

Tactic:
refine HenselianLocalRing.is_henselian f hf a₀ h₁ ?_

Premises used: HenselianLocalRing.is_henselian
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  IsUnit (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
contrapose :
  ∀ (h₂ : ¬IsUnit (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f))),
    ¬IsUnit (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
        ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
            (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
          HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
⊢ IsUnit (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f))

Tactic:
contrapose h₂
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ : ¬IsUnit (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f))
⊢ Eq (α := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    (0 : HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))

Tactic:
rw [← mem_nonunits_iff, ← IsLocalRing.mem_maximalIdeal, ← Ideal.Quotient.eq_zero_iff_mem] at h₂

Premises used: mem_nonunits_iff, IsLocalRing.mem_maximalIdeal, Ideal.Quotient.eq_zero_iff_mem
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  Eq (α := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    (0 : HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
rw : ¬IsUnit.{u_1} (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R)) 0
⊢ ¬IsUnit (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
      ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
          (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
        HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))

Tactic:
rw [h₂]
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  Eq (α := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    (0 : HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
⊢ ¬IsUnit.{u_1} (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R)) 0

Tactic:
not_isUnit_zero

Premises used: not_isUnit_zero
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Membership.mem (γ := Ideal R) (maximalIdeal R) (Polynomial.eval a₀ f)
h₂ :
  Eq (α := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    ((Ideal.Quotient.mk (maximalIdeal R) : (a : R) → R ⧸ maximalIdeal R)
        (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
      HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
    (0 : HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R))
⊢ ¬IsUnit.{u_1} (M := HasQuotient.Quotient R (B := Ideal R) (maximalIdeal R : Ideal R)) 0

Tactic:
exact not_isUnit_zero

Premises used: not_isUnit_zero
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
_ :
  f.Monic →
    ∀ (a₀ : R),
      Polynomial.eval a₀ f ∈ I →
        IsUnit (M := R ⧸ I)
            ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
              R ⧸ I) →
          ∃ (a : R), f.IsRoot a ∧ a - a₀ ∈ I
⊢ f.Monic →
    ∀ (a₀ : R),
      Polynomial.eval a₀ f ∈ I →
        IsUnit (M := R ⧸ I)
            ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) :
              R ⧸ I) →
          ∃ (a : R), f.IsRoot a ∧ a - a₀ ∈ I

Tactic:
_
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
n : ℕ
⊢ c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ

Tactic:
simp only [c]
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
⊢ ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ

Tactic:
have hc : ∀ n, c (n + 1) = c n - f.eval (c n) * (f'.eval (c n))⁻¹ʳ := by
        intro n
        simp only [c]
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
⊢ SModEq (R := R) I (c 0) a₀

Tactic:
rfl
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
⊢ SModEq (R := R) I (c (n + 1)) a₀

Tactic:
?_
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
succ : ∀ (n : ℕ) (ih : SModEq (R := R) I (c n) a₀), SModEq (R := R) I (c (n + 1)) a₀
⊢ SModEq (R := R) I (c n) a₀

Tactic:
induction n with
        | zero => rfl
        | succ n ih => ?_
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
succ : SModEq (R := R) I (c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) (a₀ + 0)
⊢ SModEq (R := R) I (c (n + 1)) a₀

Tactic:
rw [hc, sub_eq_add_neg, ← add_zero a₀]

Premises used: sub_eq_add_neg, add_zero
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
⊢ SModEq (R := R) I (c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) (a₀ + 0)

Tactic:
ih.add ?_

Premises used: SModEq.add
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
succ : SModEq (R := R) I (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) 0
⊢ SModEq (R := R) I (c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) (a₀ + 0)

Tactic:
refine ih.add ?_

Premises used: SModEq.add
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
succ : Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ ∈ I
⊢ SModEq (R := R) I (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) 0

Tactic:
rw [SModEq.zero, Ideal.neg_mem_iff]

Premises used: SModEq.zero, Ideal.neg_mem_iff
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
⊢ Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ ∈ I

Tactic:
I.mul_mem_right _ ?_

Premises used: Set.Mem, Ideal.mul_mem_right
-- Module: Mathlib.RingTheory.Henselian
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
f : R[X]
h : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ :
  IsUnit (M := R ⧸ I)
    ((Ideal.Quotient.mk I : (a : R) → R ⧸ I) (Polynomial.eval a₀ ((derivative : (a : R[X]) → R[X]) f)) : R ⧸ I)
f' : R[X] := (derivative : (a : R[X]) → R[X]) f
c : ℕ → R :=
  fun (n : ℕ) ↦
    Nat.recOn (motive := fun (x : ℕ) ↦ R) n a₀ fun (x : ℕ) (b : R) ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
n : ℕ
ih : SModEq (R := R) I (c n) a₀
succ : Polynomial.eval (c n) f ∈ I
⊢ Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ ∈ I

Tactic:
refine I.mul_mem_right _ ?_

Premises used: Ideal.mul_mem_right