awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (hypothesis)
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))
hS :
  Membership.mem (γ := Filter (WithZero (Multiplicative ℤ)))
    (𝓝
        (((Valued.v : Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) :
            (a : adicCompletion K⟮X⟯ (idealX K)) → WithZero (Multiplicative ℤ))
          a) :
      Filter (WithZero (Multiplicative ℤ)))
    S
⊢ Membership.mem (γ := Filter K⟮X⟯)
    (Filter.comap
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      (𝓝 a))
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
have hS₁ : ∃ (i : WithZero (Multiplicative ℤ)), i ≠ 0 ∧ Set.Iio i ⊆ S
Tactic: rw [ha, map_zero, WithZeroTopology.hasBasis_nhds_zero.1 S] at hS
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
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))
hS : ∃ (i : WithZero (Multiplicative ℤ)), i ≠ 0 ∧ Set.Iio i ⊆ S
⊢ Membership.mem (γ := Filter K⟮X⟯)
    (Filter.comap
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      (𝓝 a))
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
have pos : ∀ (γ : WithZero (Multiplicative ℤ)) (γ_ne_zero : γ ≠ 0) (γ_le : Set.Iio γ ⊆ S),
  Membership.mem (γ := Filter K⟮X⟯)
    (Filter.comap
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      (𝓝 a))
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
Tactic: obtain ⟨γ, γ_ne_zero, γ_le⟩ := hS
Cleared hypotheses: hS
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
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
⊢ Membership.mem (γ := Filter K⟮X⟯)
    (Filter.comap
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      (𝓝 a))
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
have h : {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} ∈
    𝓝 a ∧
  Subset.{u_2} (α := Set K⟮X⟯)
    (Set.preimage (β := adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      {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})
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
Tactic: use {t | Valued.v t < γ}
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
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
⊢ {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} ∈
      𝓝 a ∧
    Subset.{u_2} (α := Set K⟮X⟯)
      (Set.preimage (β := adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
        (fun (x : K⟮X⟯) ↦
          (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                  (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
                x) :
            UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
        {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})
      ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
have h : {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} ∈
  𝓝 a
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
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
⊢ {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} ∈
    𝓝 a
have h : ∃ (γ_1 :
  (MonoidWithZeroHom.ValueGroup₀ (F :=
      Valuation (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)) (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
  Subset.{u_2} (α := Set (adicCompletion (R := K[X]) K⟮X⟯ (idealX K)))
    {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_1 : MonoidWithZeroHom.ValueGroup₀ 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: rw [ha, this]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
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
⊢ Subset.{u_2} (α := Set K⟮X⟯)
    (Set.preimage (β := adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      {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})
    ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' S)
have h : ∀ (a : K⟮X⟯),
  Membership.mem (γ := Set K⟮X⟯)
      (Set.preimage (β := adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
        (fun (x : K⟮X⟯) ↦
          (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                  (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
                x) :
            UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
        {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})
      a →
    Membership.mem (γ := Set K⟮X⟯) ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' Set.Iio γ) a
Tactic: refine Set.Subset.trans (fun a _ ↦ ?_) (Set.preimage_mono γ_le)
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
a_1 : 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_1 = 0
S : Set (WithZero (Multiplicative ℤ))
γ : WithZero (Multiplicative ℤ)
γ_ne_zero : γ ≠ 0
γ_le : Set.Iio γ ⊆ S
a : K⟮X⟯
h :
  Membership.mem (γ := Set K⟮X⟯)
    (Set.preimage (β := adicCompletion (R := K[X]) K⟮X⟯ (idealX K))
      (fun (x : K⟮X⟯) ↦
        (↑((RingEquiv.symm (WithVal.equiv (valuation (R := K[X]) K⟮X⟯ (idealX K))) :
                (a : K⟮X⟯) → WithVal (valuation K⟮X⟯ (idealX K)))
              x) :
          UniformSpace.Completion (WithVal (valuation K⟮X⟯ (idealX K)))))
      {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})
    a
⊢ Membership.mem (γ := Set K⟮X⟯) ((⇑Valued.v : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) ⁻¹' Set.Iio γ) a
have h : ((Valued.v : Valuation (UniformSpace.Completion K⟮X⟯) (WithZero (Multiplicative ℤ))) :
      (a : UniformSpace.Completion K⟮X⟯) → WithZero (Multiplicative ℤ))
    (↑a : UniformSpace.Completion K⟮X⟯) <
  γ
Tactic: rw [Set.mem_preimage, Set.mem_Iio, ← Valued.valuedCompletion_apply a]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
this : UniformSpace (space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg) :=
  uniformStruct ratfuncAdicComplPkg
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
          (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
      WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
      WithZero (Multiplicative ℤ))
have rw : Eq (α := WithZero (Multiplicative ℤ))
  (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
        (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
      (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
    WithZero (Multiplicative ℤ))
  (Function.comp (β := space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg)
      (IsDenseInducing.extend sorry (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ)))
      (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg) f :
    WithZero (Multiplicative ℤ))
Tactic: rw [← valuation_LaurentSeries_equal_extension, ← compare_comp_eq_compare ratfuncAdicComplPkg _]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
this : UniformSpace (space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg) :=
  uniformStruct ratfuncAdicComplPkg
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
          (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
      WithZero (Multiplicative ℤ))
    (Function.comp (β := space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg)
        (IsDenseInducing.extend ⋯ (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ)))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg) f :
      WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
          (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
      WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
      WithZero (Multiplicative ℤ))
have cont_f : Continuous.{u_2, 0} (X := WithVal (R := K⟮X⟯) (polynomialValuationX K)) (Y := WithZero (Multiplicative ℤ))
  (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ))
Tactic: rw [← valuation_LaurentSeries_equal_extension, ← compare_comp_eq_compare ratfuncAdicComplPkg _]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
this : UniformSpace (space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg) :=
  uniformStruct ratfuncAdicComplPkg
rw :
  Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
          (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
      WithZero (Multiplicative ℤ))
    (Function.comp (β := space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg)
        (IsDenseInducing.extend ⋯ (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ)))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg) f :
      WithZero (Multiplicative ℤ))
cont_f :
  Continuous.{u_2, 0} (X := WithVal (R := K⟮X⟯) (polynomialValuationX K)) (Y := WithZero (Multiplicative ℤ))
    (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ))
⊢ Eq (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation (RatFuncAdicCompl K) (WithZero (Multiplicative ℤ))) :
          (a : RatFuncAdicCompl K) → WithZero (Multiplicative ℤ))
        (AbstractCompletion.compare (LaurentSeriesPkg K) ratfuncAdicComplPkg f) :
      WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
      WithZero (Multiplicative ℤ))
have a : ∀ (a : space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg),
  Tendsto (β := WithZero (Multiplicative ℤ))
    (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ))
    (Filter.comap (AbstractCompletion.coe ratfuncAdicComplPkg) (𝓝 a))
    (𝓝 (IsDenseInducing.extend sorry (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ)) a))
Tactic: rw [← valuation_LaurentSeries_equal_extension, ← compare_comp_eq_compare ratfuncAdicComplPkg _]
-- Lean 4 proof step from Mathlib.RingTheory.LaurentSeries (goal)
Goal:
K : Type u_2
inst : Field K
f : K⸨X⸩
this : UniformSpace (space (α := WithVal (R := K⟮X⟯) (polynomialValuationX K)) ratfuncAdicComplPkg) :=
  uniformStruct ratfuncAdicComplPkg
⊢ Continuous.{u_2, 0} (X := WithVal (R := K⟮X⟯) (polynomialValuationX K)) (Y := WithZero (Multiplicative ℤ))
    (⇑Valued.v : (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ))
have cont_f : ∀ (x : WithZero (Multiplicative ℤ)),
  ∃ (a : WithVal (R := K⟮X⟯) (polynomialValuationX K)),
    ((Valued.v : Valuation (WithVal (R := K⟮X⟯) (polynomialValuationX K)) (WithZero (Multiplicative ℤ))) :
          (a : WithVal (polynomialValuationX K)) → WithZero (Multiplicative ℤ))
        a =
      x
Tactic: refine Valued.continuous_valuation_of_surjective (fun x ↦ ?_)