awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
h :
  Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
      {P : K⸨X⸩ × K⸨X⸩ |
        ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (P.2 - P.1) <
          (↑γ : WithZero (Multiplicative ℤ))} ∧
    ∀ (y : K⸨X⸩ × K⸨X⸩),
      Membership.mem (γ := Set (K⸨X⸩ × K⸨X⸩))
          {P : K⸨X⸩ × K⸨X⸩ |
            ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
                (P.2 - P.1) <
              (↑γ : WithZero (Multiplicative ℤ))}
          y →
        (y.1.coeff d, y.2.coeff d) ∈ S
⊢ ∃ (v : Set (K⸨X⸩ × K⸨X⸩)),
    Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩) v ∧ ∀ y ∈ v, (y.1.coeff d, y.2.coeff d) ∈ S

Tactic:
use {P | Valued.v (P.snd - P.fst) < ↑γ}
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
⊢ Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
      {P : K⸨X⸩ × K⸨X⸩ |
        ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (P.2 - P.1) <
          (↑γ : WithZero (Multiplicative ℤ))} ∧
    ∀ (y : K⸨X⸩ × K⸨X⸩),
      Membership.mem (γ := Set (K⸨X⸩ × K⸨X⸩))
          {P : K⸨X⸩ × K⸨X⸩ |
            ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
                (P.2 - P.1) <
              (↑γ : WithZero (Multiplicative ℤ))}
          y →
        (y.1.coeff d, y.2.coeff d) ∈ S

Tactic:
⟨?_, fun _ hP ↦ ?_⟩

Premises used: And.intro
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
h :
  Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
    {P : K⸨X⸩ × K⸨X⸩ |
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          (P.2 - P.1) <
        (↑γ : WithZero (Multiplicative ℤ))}
h₁ :
  ∀ (x : K⸨X⸩ × K⸨X⸩)
    (hP :
      Membership.mem (γ := Set (K⸨X⸩ × K⸨X⸩))
        {P : K⸨X⸩ × K⸨X⸩ |
          ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
              (P.2 - P.1) <
            (↑γ : WithZero (Multiplicative ℤ))}
        x),
    (x.1.coeff d, x.2.coeff d) ∈ S
⊢ Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
      {P : K⸨X⸩ × K⸨X⸩ |
        ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (P.2 - P.1) <
          (↑γ : WithZero (Multiplicative ℤ))} ∧
    ∀ (y : K⸨X⸩ × K⸨X⸩),
      Membership.mem (γ := Set (K⸨X⸩ × K⸨X⸩))
          {P : K⸨X⸩ × K⸨X⸩ |
            ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
                (P.2 - P.1) <
              (↑γ : WithZero (Multiplicative ℤ))}
          y →
        (y.1.coeff d, y.2.coeff d) ∈ S

Tactic:
refine ⟨?_, fun _ hP ↦ ?_⟩
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
h :
  ∀ (x : K⸨X⸩)
    (hx :
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) x =
        (↑γ : WithZero (Multiplicative ℤ))),
    Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
      {P : K⸨X⸩ × K⸨X⸩ |
        ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (P.2 - P.1) <
          (↑γ : WithZero (Multiplicative ℤ))}
⊢ Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
    {P : K⸨X⸩ × K⸨X⸩ |
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          (P.2 - P.1) <
        (↑γ : WithZero (Multiplicative ℤ))}

Tactic:
obtain ⟨x, hx⟩ := LaurentSeries.valuation_surjective K γ

Premises used: Units.val, LaurentSeries.valuation_surjective
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
x : K⸨X⸩
hx :
  ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) x =
    (↑γ : WithZero (Multiplicative ℤ))
⊢ Ne (α :=
    MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
      (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (((Valuation.restrict Valued.v :
            Valuation K⸨X⸩
              (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
          (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        x :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (0 :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))

Tactic:
have : Valued.v.restrict x ≠ 0 := fun h ↦ NeZero.ne γ.1 <|
      hx ▸ MonoidWithZeroHom.ValueGroup₀.restrict₀_eq_zero_iff.1 h
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
x : K⸨X⸩
hx :
  ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) x =
    (↑γ : WithZero (Multiplicative ℤ))
this :
  Ne (α :=
    MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
      (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (((Valuation.restrict Valued.v :
            Valuation K⸨X⸩
              (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
          (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        x :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (0 :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
h :
  Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
    {P : K⸨X⸩ × K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (P.2 - P.1) :
          WithZero (Multiplicative ℤ))
        ((MonoidWithZeroHom.ValueGroup₀.embedding :
              (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (((MonoidWithZeroHom.ValueGroup₀.restrict₀ Valued.v :
                    K⸨X⸩ →*₀
                      MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v) :
                  (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ Valued.v)
                x :
              MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v) :
          WithZero (Multiplicative ℤ))}
⊢ Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
    {P : K⸨X⸩ × K⸨X⸩ |
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          (P.2 - P.1) <
        (↑γ : WithZero (Multiplicative ℤ))}

Tactic:
rw [← hx, ← MonoidWithZeroHom.ValueGroup₀.embedding_restrict₀]

Premises used: MonoidWithZeroHom.ValueGroup₀.embedding_restrict₀
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
x : K⸨X⸩
hx :
  ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) x =
    (↑γ : WithZero (Multiplicative ℤ))
this :
  Ne (α :=
    MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
      (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (((Valuation.restrict Valued.v :
            Valuation K⸨X⸩
              (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
          (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        x :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (0 :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
⊢ True

Tactic:
tauto
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
x : K⸨X⸩
hx :
  ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) x =
    (↑γ : WithZero (Multiplicative ℤ))
this :
  Ne (α :=
    MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
      (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (((Valuation.restrict Valued.v :
            Valuation K⸨X⸩
              (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
          (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        x :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
    (0 :
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
⊢ Membership.mem (γ := Filter (K⸨X⸩ × K⸨X⸩)) (uniformity K⸨X⸩)
    {P : K⸨X⸩ × K⸨X⸩ |
      LT.lt (α :=
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        (((Valuation.restrict Valued.v :
                Valuation K⸨X⸩
                  (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                    (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
              (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
            (P.2 - P.1) :
          MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
            (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
        (((MonoidWithZeroHom.ValueGroup₀.restrict₀ Valued.v :
                K⸨X⸩ →*₀ MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v) :
              (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ Valued.v)
            x :
          MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)}

Tactic:
exact (Valued.hasBasis_uniformity K⸨X⸩ ℤᵐ⁰).mem_of_mem
      (i := Units.mk0 (Valued.v.restrict x) this) (by tauto)

Premises used: Int, Multiplicative, WithZero, Valued.hasBasis_uniformity, LaurentSeries, Filter.HasBasis.mem_of_mem
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
uK : UniformSpace K
d : ℤ
S : Set (K × K)
hS : S ∈ uniformity K
γ : (WithZero (Multiplicative ℤ))ˣ := Units.mk0 (exp (-(d + 1))) ⋯
x : K⸨X⸩ × K⸨X⸩
hP :
  Membership.mem (γ := Set (K⸨X⸩ × K⸨X⸩))
    {P : K⸨X⸩ × K⸨X⸩ |
      ((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
          (P.2 - P.1) <
        (↑γ : WithZero (Multiplicative ℤ))}
    x
⊢ (x.1.coeff d, x.2.coeff d) ∈ S

Tactic:
simpa [eq_coeff_of_valuation_sub_lt K hP.le (lt_add_one _)] using mem_uniformity_of_eq hS rfl

Premises used: lt_add_one, LT.lt.le, mem_uniformity_of_eq, LaurentSeries.eq_coeff_of_valuation_sub_lt, rfl
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
D : ℤ
⊢ Tendsto (β := K) (fun (f : K⸨X⸩) ↦ f.coeff D) ℱ (𝓟 {coeff hℱ D})

Tactic:
let _ : UniformSpace K := ⊥
  le_of_eq <| DiscreteUniformity.eq_pure_cauchyConst
    (hℱ.map (uniformContinuous_coeff D)) ▸ (principal_singleton _).symm

Premises used: Cauchy.map, Eq.rec, Bot.bot, Filter.Tendsto, Filter.principal_singleton, DiscreteUniformity.eq_pure_cauchyConst, LaurentSeries.uniformContinuous_coeff, le_of_eq, Eq.symm, DiscreteUniformity.cauchyConst
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
⊢ True

Tactic:
tauto
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
obtain :
  ∀ (S : Set K⸨X⸩) (hS : S ∈ ℱ) (T : Set K⸨X⸩) (hT : T ∈ ℱ) (H : S ×ˢ T ⊆ entourage),
    ∃ (N : ℤ), ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
⊢ ∃ (N : ℤ), ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0

Tactic:
obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := mem_prod_iff.mp <| Filter.le_def.mp hℱ.2 entourage
    <| (Valued.hasBasis_uniformity K⸨X⸩ ℤᵐ⁰).mem_of_mem (i := ζ) (by tauto)

Premises used: Int, Multiplicative, And.right, WithZero, Iff.mp, Valued.hasBasis_uniformity, LaurentSeries, Filter.mem_prod_iff, Filter.HasBasis.mem_of_mem, Filter.le_def
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
S : Set K⸨X⸩
hS : S ∈ ℱ
T : Set K⸨X⸩
hT : T ∈ ℱ
H : S ×ˢ T ⊆ entourage
obtain : ∀ (f : K⸨X⸩) (hf : f ∈ S ∩ T), ∃ (N : ℤ), ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
⊢ ∃ (N : ℤ), ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0

Tactic:
obtain ⟨f, hf⟩ := forall_mem_nonempty_iff_neBot.mpr hℱ.1 (S ∩ T) (inter_mem_iff.mpr ⟨hS, hT⟩)

Premises used: Iff.mpr, Filter.forall_mem_nonempty_iff_neBot, Inter.inter, And.left, And.intro, Filter.inter_mem_iff
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
S : Set K⸨X⸩
hS : S ∈ ℱ
T : Set K⸨X⸩
hT : T ∈ ℱ
H : S ×ˢ T ⊆ entourage
f : K⸨X⸩
hf : f ∈ S ∩ T
pos :
  ∀ (hf : f = 0),
    ∃ (N : ℤ),
      ∀ (g : K⸨X⸩),
        LE.le (α := WithZero (Multiplicative ℤ))
            (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
                (g - f) :
              WithZero (Multiplicative ℤ))
            (1 : WithZero (Multiplicative ℤ)) →
          ∀ n < N, g.coeff n = 0
neg :
  ∀ (hf : ¬f = 0),
    ∃ (N : ℤ),
      ∀ (g : K⸨X⸩),
        LE.le (α := WithZero (Multiplicative ℤ))
            (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
                (g - f) :
              WithZero (Multiplicative ℤ))
            (1 : WithZero (Multiplicative ℤ)) →
          ∀ n < N, g.coeff n = 0
⊢ ∃ (N : ℤ),
    ∀ (g : K⸨X⸩),
      LE.le (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
              (g - f) :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ)) →
        ∀ n < N, g.coeff n = 0

Tactic:
by_cases hf : f = 0
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
S : Set K⸨X⸩
hS : S ∈ ℱ
T : Set K⸨X⸩
hT : T ∈ ℱ
H : S ×ˢ T ⊆ entourage
f : K⸨X⸩
hf_1 : f ∈ S ∩ T
hf : f = 0
⊢ ∃ (N : ℤ),
    ∀ (g : K⸨X⸩),
      LE.le (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
              (g - f) :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ)) →
        ∀ n < N, g.coeff n = 0

Tactic:
⟨0, fun x hg ↦ ?_⟩

Premises used: Exists.intro, OfNat.ofNat
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
entourage : Set (K⸨X⸩ × K⸨X⸩) :=
  {P : K⸨X⸩ × K⸨X⸩ |
    LT.lt (α :=
      MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
        (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (((Valuation.restrict Valued.v :
              Valuation K⸨X⸩
                (MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
                  (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))) :
            (a : K⸨X⸩) → MonoidWithZeroHom.ValueGroup₀ (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
          (P.2 - P.1) :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))
      (1 :
        MonoidWithZeroHom.ValueGroup₀ (F := K⸨X⸩ →*₀ WithZero (Multiplicative ℤ))
          (↑Valued.v : K⸨X⸩ →*₀ WithZero (Multiplicative ℤ)))}
ζ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ := Units.mk0 1 ⋯
S : Set K⸨X⸩
hS : S ∈ ℱ
T : Set K⸨X⸩
hT : T ∈ ℱ
H : S ×ˢ T ⊆ entourage
f : K⸨X⸩
hf_1 : f ∈ S ∩ T
hf : f = 0
pos :
  ∀ (x : K⸨X⸩)
    (hg :
      LE.le (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
            (x - f) :
          WithZero (Multiplicative ℤ))
        (1 : WithZero (Multiplicative ℤ))),
    ∀ n < 0, x.coeff n = 0
⊢ ∃ (N : ℤ),
    ∀ (g : K⸨X⸩),
      LE.le (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
              (g - f) :
            WithZero (Multiplicative ℤ))
          (1 : WithZero (Multiplicative ℤ)) →
        ∀ n < N, g.coeff n = 0

Tactic:
refine ⟨0, fun x hg ↦ ?_⟩