-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
a : adicCompletion (R := K[X]) K⟮X⟯ (idealX K)
this :
  ∀ (s : Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))),
    Membership.mem (γ := Filter (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))) (𝓝 0) s ↔
      ∃ (γ :
        (MonoidWithZeroHom.ValueGroup₀ (F :=
            Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
        {x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
            ((Valuation.restrict Valued.v :
                    Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                      (MonoidWithZeroHom.ValueGroup₀ (F :=
                        adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                        (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
                  (a : adicCompletion K⟮X⟯ (idealX K)) →
                    MonoidWithZeroHom.ValueGroup₀
                      (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
                x <
              (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
          s
ha : a = 0
S : Set (WithZero (Multiplicative ℤ))
γ : WithZero (Multiplicative ℤ)
γ_ne_zero : γ ≠ 0
γ_le : Set.Iio γ ⊆ S
x : adicCompletion (R := K[X]) K⟮X⟯ (idealX K)
hx :
  ((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
        (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
      x =
    γ
⊢ Subset.{u_2} (α := Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)))
    {x_1 : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
      LT.lt (α :=
        MonoidWithZeroHom.ValueGroup₀ (F := adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
        (((Valuation.restrict Valued.v :
                Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                  (MonoidWithZeroHom.ValueGroup₀ (F :=
                    adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                    (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
              (a : adicCompletion K⟮X⟯ (idealX K)) →
                MonoidWithZeroHom.ValueGroup₀
                  (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
            x_1 :
          MonoidWithZeroHom.ValueGroup₀ (F :=
            adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
            (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
        ((↑(Units.mk0
                (((Valuation.restrict Valued.v :
                      Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
                        (MonoidWithZeroHom.ValueGroup₀ (F :=
                          adicCompletion (R := K[X]) K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ))
                          (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))) :
                    (a : adicCompletion K⟮X⟯ (idealX K)) →
                      MonoidWithZeroHom.ValueGroup₀
                        (↑Valued.v : adicCompletion K⟮X⟯ (idealX K) →*₀ WithZero (Multiplicative ℤ)))
                  x)
                ⋯) :
            MonoidWithZeroHom.ValueGroup₀ (F :=
              Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v) :
          MonoidWithZeroHom.ValueGroup₀ (F :=
            Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)}
    {t : adicCompletion (R := K[X]) K⟮X⟯ (idealX K) |
      ((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
            (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
          t <
        γ}

Tactic:
simp [Units.val_mk0, Valuation.restrict_lt_iff, hx]

Premises used: Valuation.restrict_lt_iff, Units.val_mk0