awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T :
  (fun (x : K⟮X⟯ × K⟮X⟯) ↦
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
      T ⊆
    S
R : Set K⸨X⸩
hR : Membership.mem (γ := Filter K⸨X⸩) (nhds 0) R
pre_R : (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⸨X⸩ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    R
x : K⟮X⟯ × K⟮X⟯
h_1 :
  Membership.mem (γ := Set (K⟮X⟯ × K⟮X⟯))
    (Set.preimage (β := K⟮X⟯) (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1)
      {P : K⟮X⟯ |
        LT.lt (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) P :
            WithZero (Multiplicative ℤ))
          ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
              (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
            WithZero (Multiplicative ℤ))})
    x
h :
  ((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)))
      (HSub.hSub (α := K⸨X⸩) (β := K⸨X⸩) ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2 : K⸨X⸩)
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1 : K⸨X⸩)) <
    (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)
⊢ Membership.mem (γ := Set K⸨X⸩)
    {y : K⸨X⸩ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)}
    ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1)
      ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2))

Tactic:
simp only [sub_zero, Set.mem_setOf_eq]

Premises used: sub_zero, Set.mem_setOf_eq
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T :
  (fun (x : K⟮X⟯ × K⟮X⟯) ↦
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
      T ⊆
    S
R : Set K⸨X⸩
hR : Membership.mem (γ := Filter K⸨X⸩) (nhds 0) R
pre_R : (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⸨X⸩ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    R
x : K⟮X⟯ × K⟮X⟯
h_1 :
  Membership.mem (γ := Set (K⟮X⟯ × K⟮X⟯))
    (Set.preimage (β := K⟮X⟯) (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1)
      {P : K⟮X⟯ |
        LT.lt (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) P :
            WithZero (Multiplicative ℤ))
          ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
              (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
            WithZero (Multiplicative ℤ))})
    x
h :
  LT.lt (α := WithZero (Multiplicative ℤ))
    (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) (x.2 - x.1)) :
      WithZero (Multiplicative ℤ))
    ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
      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)))
      (HSub.hSub (α := K⸨X⸩) (β := K⸨X⸩) ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2 : K⸨X⸩)
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1 : K⸨X⸩)) <
    (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)

Tactic:
rw [← map_sub, Valuation.restrict_lt_iff_lt_embedding]

Premises used: map_sub, Valuation.restrict_lt_iff_lt_embedding
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
T : Set (K⸨X⸩ × K⸨X⸩)
pre_T :
  (fun (x : K⟮X⟯ × K⟮X⟯) ↦
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
      T ⊆
    S
R : Set K⸨X⸩
hR : Membership.mem (γ := Filter K⸨X⸩) (nhds 0) R
pre_R : (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' R ⊆ T
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⸨X⸩ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    R
x : K⟮X⟯ × K⟮X⟯
h_1 :
  Membership.mem (γ := Set (K⟮X⟯ × K⟮X⟯))
    (Set.preimage (β := K⟮X⟯) (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1)
      {P : K⟮X⟯ |
        LT.lt (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) : (a : K⟮X⟯) → WithZero (Multiplicative ℤ)) P :
            WithZero (Multiplicative ℤ))
          ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
              (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
            WithZero (Multiplicative ℤ))})
    x
h h₁ h₂ h₃ :
  LT.lt (α := WithZero (Multiplicative ℤ))
    (((polynomialValuationX K : Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) :
          (a : K⟮X⟯) → WithZero (Multiplicative ℤ))
        (x.2 - x.1) :
      WithZero (Multiplicative ℤ))
    ((embedding :
          (a : MonoidWithZeroHom.ValueGroup₀ (valuation K⸨X⸩ (PowerSeries.idealX K))) → WithZero (Multiplicative ℤ))
        (↑d : MonoidWithZeroHom.ValueGroup₀ (valuation K⸨X⸩ (PowerSeries.idealX K))) :
      WithZero (Multiplicative ℤ))
⊢ LT.lt (α := WithZero (Multiplicative ℤ))
    (((valuation K⸨X⸩ (PowerSeries.idealX K) : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) :
          (a : K⸨X⸩) → WithZero (Multiplicative ℤ))
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) (x.2 - x.1)) :
      WithZero (Multiplicative ℤ))
    ((embedding :
          (a : MonoidWithZeroHom.ValueGroup₀ (valuation K⸨X⸩ (PowerSeries.idealX K))) → WithZero (Multiplicative ℤ))
        (↑d : MonoidWithZeroHom.ValueGroup₀ (valuation K⸨X⸩ (PowerSeries.idealX K))) :
      WithZero (Multiplicative ℤ))

Tactic:
rwa [← valuation_eq_LaurentSeries_valuation]

Premises used: RatFunc.valuation_eq_LaurentSeries_valuation
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
h :
  ∀ (d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ)
    (hd :
      {y : K⟮X⟯ |
          ((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)))
              (y - 0) <
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
        w),
    ∃ (t : Set (K⸨X⸩ × K⸨X⸩)),
      (∃ (t_1 : Set K⸨X⸩),
          Membership.mem (γ := Filter K⸨X⸩) (nhds 0) t_1 ∧ (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
        (fun (x : K⟮X⟯ × K⟮X⟯) ↦
              ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
            t ⊆
          S
⊢ ∃ (t : Set (K⸨X⸩ × K⸨X⸩)),
    (∃ (t_1 : Set K⸨X⸩),
        Membership.mem (γ := Filter K⸨X⸩) (nhds 0) t_1 ∧ (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
      (fun (x : K⟮X⟯ × K⟮X⟯) ↦
            ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
          t ⊆
        S

Tactic:
obtain ⟨d, hd⟩ := Valued.mem_nhds.mp hT

Premises used: Valued.mem_nhds, Iff.mp
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
⊢ let X : Set K⸨X⸩ :=
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))};
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}

Tactic:
set X := {f : K⸨X⸩ | Valued.v f < embedding d.1} with X_def

Premises used: Units.val, MonoidWithZeroHom.ValueGroup₀.embedding, DFunLike.coe
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
X : Set K⸨X⸩ :=
  {f : K⸨X⸩ |
    LT.lt (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
        WithZero (Multiplicative ℤ))
      ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
        WithZero (Multiplicative ℤ))}
X_def :
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}
⊢ ∃ (t : Set (K⸨X⸩ × K⸨X⸩)),
    (∃ (t_1 : Set K⸨X⸩),
        Membership.mem (γ := Filter K⸨X⸩) (nhds 0) t_1 ∧ (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
      (fun (x : K⟮X⟯ × K⟮X⟯) ↦
            ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
          t ⊆
        S

Tactic:
⟨(fun x : K⸨X⸩ × K⸨X⸩ ↦ x.snd - x.fst) ⁻¹' X, ⟨X, ?_⟩, ?_⟩

Premises used: HSub.hSub, Prod.fst, Set.preimage, And.intro, Exists.intro, Prod.snd
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
X : Set K⸨X⸩ :=
  {f : K⸨X⸩ |
    LT.lt (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
        WithZero (Multiplicative ℤ))
      ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
        WithZero (Multiplicative ℤ))}
X_def :
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}
h :
  Membership.mem (γ := Filter K⸨X⸩) (nhds 0) X ∧
    Subset (α := Set (K⸨X⸩ × K⸨X⸩)) ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)
      ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)
h₁ :
  Set.preimage (β := K⸨X⸩ × K⸨X⸩)
      (fun (x : K⟮X⟯ × K⟮X⟯) ↦
        ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2))
      ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X) ⊆
    S
⊢ ∃ (t : Set (K⸨X⸩ × K⸨X⸩)),
    (∃ (t_1 : Set K⸨X⸩),
        Membership.mem (γ := Filter K⸨X⸩) (nhds 0) t_1 ∧ (fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧
      (fun (x : K⟮X⟯ × K⟮X⟯) ↦
            ((algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.1, (algebraMap K⟮X⟯ K⸨X⸩ : (a : K⟮X⟯) → K⸨X⸩) x.2)) ⁻¹'
          t ⊆
        S

Tactic:
refine ⟨(fun x : K⸨X⸩ × K⸨X⸩ ↦ x.snd - x.fst) ⁻¹' X, ⟨X, ?_⟩, ?_⟩
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
X : Set K⸨X⸩ :=
  {f : K⸨X⸩ |
    LT.lt (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
        WithZero (Multiplicative ℤ))
      ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
        WithZero (Multiplicative ℤ))}
X_def :
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}
⊢ Membership.mem (γ := Filter K⸨X⸩) (nhds 0) X ∧
    Subset (α := Set (K⸨X⸩ × K⸨X⸩)) ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)
      ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)

Tactic:
⟨?_, Set.Subset.refl _⟩

Premises used: Set.Subset.refl, And.intro
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
X : Set K⸨X⸩ :=
  {f : K⸨X⸩ |
    LT.lt (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
        WithZero (Multiplicative ℤ))
      ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
        WithZero (Multiplicative ℤ))}
X_def :
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}
h : Membership.mem (γ := Filter K⸨X⸩) (nhds 0) X
⊢ Membership.mem (γ := Filter K⸨X⸩) (nhds 0) X ∧
    Subset (α := Set (K⸨X⸩ × K⸨X⸩)) ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)
      ((fun (x : K⸨X⸩ × K⸨X⸩) ↦ x.2 - x.1) ⁻¹' X)

Tactic:
refine ⟨?_, Set.Subset.refl _⟩
-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
S : Set (K⟮X⟯ × K⟮X⟯)
w : Set K⟮X⟯
hT : Membership.mem (γ := Filter K⟮X⟯) (nhds 0) w
pre_T : (fun (x : K⟮X⟯ × K⟮X⟯) ↦ x.2 - x.1) ⁻¹' w ⊆ S
d : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⟮X⟯ (WithZero (Multiplicative ℤ))) Valued.v)ˣ
hd :
  {y : K⟮X⟯ |
      ((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)))
          (y - 0) <
        (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v)} ⊆
    w
X : Set K⸨X⸩ :=
  {f : K⸨X⸩ |
    LT.lt (α := WithZero (Multiplicative ℤ))
      (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
        WithZero (Multiplicative ℤ))
      ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
          (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
        WithZero (Multiplicative ℤ))}
X_def :
  X =
    {f : K⸨X⸩ |
      LT.lt (α := WithZero (Multiplicative ℤ))
        (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) f :
          WithZero (Multiplicative ℤ))
        ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
            (↑d : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
          WithZero (Multiplicative ℤ))}
h :
  ∃ (γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) Valued.v)ˣ),
    {y : K⸨X⸩ |
        LT.lt (α := WithZero (Multiplicative ℤ))
          (((Valued.v : Valuation K⸨X⸩ (WithZero (Multiplicative ℤ))) : (a : K⸨X⸩) → WithZero (Multiplicative ℤ)) y :
            WithZero (Multiplicative ℤ))
          ((embedding : (a : MonoidWithZeroHom.ValueGroup₀ Valued.v) → WithZero (Multiplicative ℤ))
              (↑γ : MonoidWithZeroHom.ValueGroup₀ Valued.v) :
            WithZero (Multiplicative ℤ))} ⊆
      X
⊢ Membership.mem (γ := Filter K⸨X⸩) (nhds 0) X

Tactic:
simp only [Valued.mem_nhds, sub_zero, Valuation.restrict_lt_iff_lt_embedding]

Premises used: sub_zero, Valuation.restrict_lt_iff_lt_embedding, Valued.mem_nhds