-- Mathlib/RingTheory/Henselian.lean
theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [IsLocalRing R] :
    TFAE
      [HenselianLocalRing R,
        ∀ f : R[X], f.Monic → ∀ a₀ : ResidueField R, aeval a₀ f = 0 →
          aeval a₀ (derivative f) ≠ 0 → ∃ a : R, f.IsRoot a ∧ residue R a = a₀,
        ∀ {K : Type u} [Field K],
          ∀ (φ : R →+* K), Surjective φ → ∀ f : R[X], f.Monic → ∀ a₀ : K,
            f.eval₂ φ a₀ = 0 → f.derivative.eval₂ φ a₀ ≠ 0 → ∃ a : R, f.IsRoot a ∧ φ a = a₀]

Proof state:
case is_henselian
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
x✝ :
  ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : IsUnit (Polynomial.eval a₀ (derivative f))
H :
  (aeval ((residue R) a₀)) f = 0 →
    (aeval ((residue R) a₀)) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = (residue R) a₀
aux : Polynomial.eval a₀ (derivative f) ∈ nonunits R → False
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalIdeal R

Tactic:
simp only [aeval_def, ResidueField.algebraMap_eq, eval₂_at_apply, ←
  Ideal.Quotient.eq_zero_iff_mem, ← IsLocalRing.mem_maximalIdeal] at H h₁ aux

Result:
case is_henselian
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
x✝ :
  ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
f : R[X]
hf : f.Monic
a₀ : R
h₂ : IsUnit (Polynomial.eval a₀ (derivative f))
H :
  (residue R) (Polynomial.eval a₀ f) = 0 →
    (residue R) (Polynomial.eval a₀ (derivative f)) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = (residue R) a₀
h₁ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ f) = 0
aux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0 → False
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalIdeal R

Proof state:
case is_henselian.intro.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
x✝ :
  ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
f : R[X]
hf : f.Monic
a₀ : R
h₂ : IsUnit (Polynomial.eval a₀ (derivative f))
H :
  (residue R) (Polynomial.eval a₀ f) = 0 →
    (residue R) (Polynomial.eval a₀ (derivative f)) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = (residue R) a₀
h₁ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ f) = 0
aux : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0 → False
a : R
ha₁ : f.IsRoot a
ha₂ : (residue R) a = (residue R) a₀
⊢ (Ideal.Quotient.mk (maximalIdeal R)) (a - a₀) = 0

Tactic:
rwa [← sub_eq_zero, ← RingHom.map_sub] at ha₂

Result:
no goals

Proof state:
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
tfae_2_to_1 :
  (∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R),
          (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀) →
    HenselianLocalRing R
tfae_1_to_3 :
  HenselianLocalRing R →
    ∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀
⊢ [HenselianLocalRing R,
      ∀ (f : R[X]),
        f.Monic →
          ∀ (a₀ : ResidueField R),
            (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀,
      ∀ {K : Type u} [inst : Field K] (φ : R →+* K),
        Surjective ⇑φ →
          ∀ (f : R[X]),
            f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀].TFAE

Tactic:
tfae_finish

Result:
no goals

Proof state:
case is_henselian
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
x✝ :
  ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
H : ∀ (f : R[X]),
  f.Monic →
    ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀ :=
  x✝
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : IsUnit (Polynomial.eval a₀ (derivative f))
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalIdeal R

Tactic:
specialize H f hf (residue R a₀)

Result:
case is_henselian
R : Type u
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
tfae_3_to_2 :
  (∀ {K : Type u} [inst : Field K] (φ : R →+* K),
      Surjective ⇑φ →
        ∀ (f : R[X]),
          f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →
    ∀ (f : R[X]),
      f.Monic →
        ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
x✝ :
  ∀ (f : R[X]),
    f.Monic →
      ∀ (a₀ : ResidueField R), (aeval a₀) f = 0 → (aeval a₀) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = a₀
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : IsUnit (Polynomial.eval a₀ (derivative f))
H :
  (aeval ((residue R) a₀)) f = 0 →
    (aeval ((residue R) a₀)) (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ (residue R) a = (residue R) a₀
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ maximalIdeal R