-- Module: Mathlib.NumberTheory.Padics.Hensel
Goal:
p : ℕ
inst : Fact (Nat.Prime p)
R : Type u_1
inst_1 : CommSemiring R
inst_2 : Algebra R ℤ_[p]
F : Polynomial R
a : ℤ_[p]
ha :
  Eq (α := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p]) (0 : ℤ_[p])
z' : ℤ_[p]
hz' :
  Eq (α := ℤ_[p]) (((Polynomial.aeval z' : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p])
    (0 : ℤ_[p])
hnormz' :
  ‖z' - a‖ <
    norm (E := ℤ_[p])
      (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
          ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
        ℤ_[p])
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq :
  ((Polynomial.aeval (a + h) : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F =
    HAdd.hAdd (α := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p])
        (HMul.hMul (α := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p])
          h) +
      q * HPow.hPow (β := ℕ) h 2
rw :
  HAdd.hAdd (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        (q * h) *
      h =
    HAdd.hAdd (α := ℤ_[p]) (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F : ℤ_[p])
        (HMul.hMul (α := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p])
          h) +
      q * HPow.hPow (β := ℕ) h 2
rw₁ :
  HAdd.hAdd (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        (q * h) *
      h =
    0 +
        HMul.hMul (α := ℤ_[p])
          (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
              ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
            ℤ_[p])
          h +
      q * HPow.hPow (β := ℕ) h 2
rw₂ :
  HAdd.hAdd (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        (q * h) *
      h =
    HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * HPow.hPow (β := ℕ) h 2
rw₃ :
  HAdd.hAdd (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        (q * h) *
      h =
    HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * (h * h)
rw₄ :
  HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * h * h =
    HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * (h * h)
rw₅ rw₆ :
  HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * (h * h) =
    HMul.hMul (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        h +
      q * (h * h)
⊢ HAdd.hAdd (α := ℤ_[p])
        (((Polynomial.aeval a : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p])
            ((Polynomial.derivative : (a : Polynomial R) → Polynomial R) F) :
          ℤ_[p])
        (q * h) *
      h =
    ((Polynomial.aeval (a + h) : Polynomial R →ₐ[R] ℤ_[p]) : (a : Polynomial R) → ℤ_[p]) F

Tactic:
rw [hq, ha, zero_add, sq, right_distrib, mul_assoc]

Premises used: mul_assoc, sq, zero_add, right_distrib