awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
r' : K
hr' : ¬r' = 0
hr_1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
r : K
hr :
  Ne (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
    (0 : ValueGroupWithZero K)
hrs :
  {x : K |
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)} ⊆
    s
hr1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
⊢ ∃ (r' : K),
    r' ≠ 0 ∧
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
          (1 : ValueGroupWithZero K) ∧
        {x : K |
            LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
              ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)} ⊆
          s
have hr₁ : ¬r = 0
Tactic: simp only [ne_eq, map_eq_zero] at hr
Cleared hypotheses: hr
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
r' : K
hr' : ¬r' = 0
hr_1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
r : K
hrs :
  {x : K |
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)} ⊆
    s
hr1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
hr : ¬r = 0
⊢ ∃ (r' : K),
    r' ≠ 0 ∧
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
          (1 : ValueGroupWithZero K) ∧
        {x : K |
            LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
              ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)} ⊆
          s
have inl : ∀ (x : K)
  (hx :
    Membership.mem (γ := Set K)
      {x : K |
        LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
          ((valuation K : (a : K) → ValueGroupWithZero K) (HPow.hPow (β := ℕ) r 2) : ValueGroupWithZero K)}
      x),
  Membership.mem (γ := Set K)
    {x : K |
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)}
    x
Tactic: refine ⟨r ^ 2, by simpa using hr, by simpa [pow_two], fun x hx ↦ hrs ?_⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
r' : K
hr' : ¬r' = 0
hr_1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
r : K
hrs :
  {x : K |
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)} ⊆
    s
hr1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
hr : ¬r = 0
x : K
hx :
  Membership.mem (γ := Set K)
    {x : K |
      LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) (HPow.hPow (β := ℕ) r 2) : ValueGroupWithZero K)}
    x
⊢ Membership.mem (γ := Set K)
    {x : K |
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)}
    x
have hx₁ : (valuation K : (a : K) → ValueGroupWithZero K) x ≤
  HPow.hPow (α := ValueGroupWithZero K) (β := ℕ)
    ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K) 2
Tactic: simp only [map_pow, Set.mem_setOf_eq] at hx ⊢
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
r : ValueGroupWithZero K
hr_1 : r ≠ 0
hrs : {x : K | (valuation K : (a : K) → ValueGroupWithZero K) x < r} ⊆ s
r' : K
hr' : ¬r' = 0
hr :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
hr1 : 1 ≤ r
⊢ ∃ (r' : K),
    r' ≠ 0 ∧
      LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)
          (1 : ValueGroupWithZero K) ∧
        {x : K |
            LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
              ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)} ⊆
          s
have inr : Subset.{u_1} (α := Set K)
  {x : K |
    LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
      ((valuation K : (a : K) → ValueGroupWithZero K) r' : ValueGroupWithZero K)}
  {x : K | (valuation K : (a : K) → ValueGroupWithZero K) x < r}
Tactic: refine ⟨r', hr', hr, .trans ?_ hrs⟩
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
⊢ IsCompact.{u_1} (X := K)
    {x : K |
      LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) γ : ValueGroupWithZero K)}
have neg : ∀ (r : K) (hr : r ≠ 0)
  (hr1 :
    LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
      (1 : ValueGroupWithZero K))
  (H :
    {x : K |
        LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
          ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)} ⊆
      s),
  IsCompact.{u_1} (X := K)
    {x : K |
      LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) γ : ValueGroupWithZero K)}
Tactic: obtain ⟨r, hr, hr1, H⟩ :
      ∃ r', r' ≠ 0 ∧ valuation K r' < 1 ∧ { x | valuation K x ≤ valuation K r' } ⊆ s := by
    obtain ⟨r, hr, hrs⟩ := (IsValuativeTopology.hasBasis_nhds_zero' K).mem_iff.mp hs
    obtain ⟨r', hr', hr⟩ := Valuation.IsNontrivial.exists_lt_one (v := valuation K)
    simp only [ne_eq] at hr'
    obtain hr1 | hr1 := lt_or_ge r 1
    · obtain ⟨r, rfl⟩ := ValuativeRel.valuation_surjective r
      simp only [ne_eq, map_eq_zero] at hr
      refine ⟨r ^ 2, by simpa using hr, by simpa [pow_two], fun x hx ↦ hrs ?_⟩
      simp only [map_pow, Set.mem_setOf_eq] at hx ⊢
      exact hx.trans_lt (by simpa [pow_two, hr])
    · refine ⟨r', hr', hr, .trans ?_ hrs⟩
      intro x hx
      dsimp at hx ⊢
      exact hx.trans_lt (hr.trans_le hr1)
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
γ : K
hγ : ¬γ = 0
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K := ⋯
s : Set K
hs : Membership.mem (γ := Filter K) (nhds 0) s
hs' : IsCompact s
r : K
hr : r ≠ 0
hr1 :
  LT.lt (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)
    (1 : ValueGroupWithZero K)
H :
  {x : K |
      LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) r : ValueGroupWithZero K)} ⊆
    s
⊢ IsCompact.{u_1} (X := K)
    {x : K |
      LE.le (α := ValueGroupWithZero K) ((valuation K : (a : K) → ValueGroupWithZero K) x : ValueGroupWithZero K)
        ((valuation K : (a : K) → ValueGroupWithZero K) γ : ValueGroupWithZero K)}
have neg : ∀
  (H :
    {x : K |
        LE.le (α :=
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K) (↑(valuation K) : K →*₀ ValueGroupWithZero K))
          ((Valuation.restrict (valuation K) :
                (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑(valuation K) : K →*₀ ValueGroupWithZero K))
              x :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K)
              (↑(valuation K) : K →*₀ ValueGroupWithZero K))
          ((Valuation.restrict (valuation K) :
                (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑(valuation K) : K →*₀ ValueGroupWithZero K))
              r :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K)
              (↑(valuation K) : K →*₀ ValueGroupWithZero K))} ⊆
      s),
  IsCompact.{u_1} (X := K)
    {x : K |
      LE.le (α :=
        MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K) (↑(valuation K) : K →*₀ ValueGroupWithZero K))
        ((Valuation.restrict (valuation K) :
              (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑(valuation K) : K →*₀ ValueGroupWithZero K))
            x :
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K) (↑(valuation K) : K →*₀ ValueGroupWithZero K))
        ((Valuation.restrict (valuation K) :
              (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑(valuation K) : K →*₀ ValueGroupWithZero K))
            γ :
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ ValueGroupWithZero K)
            (↑(valuation K) : K →*₀ ValueGroupWithZero K))}
Tactic: simp_rw [← (valuation K).restrict_le_iff] at H ⊢
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
⊢ ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ)
have h : Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
Tactic: apply Nonempty.some
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
this : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
⊢ Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
have this₁ : IsUniformAddGroup K
Tactic: haveI := isUniformAddGroup_of_addCommGroup (G := K)
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K
⊢ Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
have h : ∀ (val : LocallyFiniteOrder (↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) Valued.v))ˣ),
  Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
Tactic: obtain ⟨_⟩ := Valued.integer.locallyFiniteOrder_units_mrange_of_isCompact_integer
    (isCompact_iff_compactSpace.mpr (inferInstance : CompactSpace 𝒪[K]))
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this : IsUniformAddGroup K
val : LocallyFiniteOrder (↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) Valued.v))ˣ
e : ↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K)) ≃*o ValueGroupWithZero K :=
  {
    toMulEquiv :=
      MulEquiv.ofBijective (F :=
        ↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K)) →* ValueGroupWithZero K)
        (Submonoid.subtype (MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K))) ⋯,
    map_le_map_iff' := ⋯ }
⊢ Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
have this₁ : Nontrivial.{u_1} (ValueGroupWithZero K)ˣ
Tactic: have : Nontrivial (ValueGroupWithZero K)ˣ := isNontrivial_iff_nontrivial_units.mp inferInstance
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : TopologicalSpace K
inst_3 : IsNonarchimedeanLocalField K
this_1 : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K
this_2 : IsUniformAddGroup K
val : LocallyFiniteOrder (↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) Valued.v))ˣ
e : ↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K)) ≃*o ValueGroupWithZero K :=
  {
    toMulEquiv :=
      MulEquiv.ofBijective (F :=
        ↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K)) →* ValueGroupWithZero K)
        (Submonoid.subtype (MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K))) ⋯,
    map_le_map_iff' := ⋯ }
this : Nontrivial.{u_1} (ValueGroupWithZero K)ˣ
⊢ Nonempty.{u_1 + 1} (ValueGroupWithZero K ≃*o WithZero (Multiplicative ℤ))
have this₁ : Nontrivial.{u_1} (↥(MonoidHom.mrange (F := Valuation K (ValueGroupWithZero K)) (valuation K)))ˣ
Tactic: have : Nontrivial (↥(MonoidHom.mrange (valuation K)))ˣ :=
    (Units.map_injective (f := e.symm.toMonoidHom) e.symm.injective).nontrivial
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : UniformSpace K
inst_3 : IsUniformAddGroup K
inst_4 : IsNonarchimedeanLocalField K
f : ℕ → ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
hf :
  ∀ {m n : ℕ},
    m ≤ n →
      f m ≡ f n [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) m)
          ⊤]
S : (n : ℕ) → Set ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)) :=
  fun (n : ℕ) ↦
    f n +ᵥ
      (↑(HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n) :
        Set ↥𝒪[K])
n : ℕ
⊢ S (n + 1) ⊆ S n
have apply : f (n + 1) +ᵥ
    (↑(HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n) :
      Set ↥𝒪[K]) ⊆
  S n
Tactic: apply (Set.vadd_set_subset_vadd_set_iff.mpr (Ideal.pow_le_pow_right n.le_succ)).trans
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : UniformSpace K
inst_3 : IsUniformAddGroup K
inst_4 : IsNonarchimedeanLocalField K
f : ℕ → ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
hf :
  ∀ {m n : ℕ},
    m ≤ n →
      f m ≡ f n [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) m)
          ⊤]
S : (n : ℕ) → Set ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)) :=
  fun (n : ℕ) ↦
    f n +ᵥ
      (↑(HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n) :
        Set ↥𝒪[K])
⊢ ∃ (L : ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))),
    ∀ (n : ℕ),
      f n ≡ L [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n)
          ⊤]
have hS : ∀ (n : ℕ), S (n + 1) ⊆ S n
Tactic: have hS n : S (n + 1) ⊆ S n := by
      apply (Set.vadd_set_subset_vadd_set_iff.mpr (Ideal.pow_le_pow_right n.le_succ)).trans
      simpa [S] using (hf n.le_succ).symm
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : UniformSpace K
inst_3 : IsUniformAddGroup K
inst_4 : IsNonarchimedeanLocalField K
f : ℕ → ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
hf :
  ∀ {m n : ℕ},
    m ≤ n →
      f m ≡ f n [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) m)
          ⊤]
S : (n : ℕ) → Set ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)) :=
  fun (n : ℕ) ↦
    f n +ᵥ
      (↑(HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n) :
        Set ↥𝒪[K])
hS : ∀ (n : ℕ), S (n + 1) ⊆ S n
⊢ ∃ (L : ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))),
    ∀ (n : ℕ),
      f n ≡ L [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n)
          ⊤]
have h : ∀ (n : ℕ), IsClosed (S n)
Tactic: have h n : IsClosed (S n) := (IsNoetherianRing.isClosed_ideal (𝓂[K] ^ n)).vadd (f n)
-- Lean 4 proof step from Mathlib.NumberTheory.LocalField.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : ValuativeRel K
inst_2 : UniformSpace K
inst_3 : IsUniformAddGroup K
inst_4 : IsNonarchimedeanLocalField K
f : ℕ → ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
hf :
  ∀ {m n : ℕ},
    m ≤ n →
      f m ≡ f n [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) m)
          ⊤]
S : (n : ℕ) → Set ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)) :=
  fun (n : ℕ) ↦
    f n +ᵥ
      (↑(HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n) :
        Set ↥𝒪[K])
hS : ∀ (n : ℕ), S (n + 1) ⊆ S n
h : ∀ (n : ℕ), IsClosed (S n)
⊢ ∃ (L : ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))),
    ∀ (n : ℕ),
      f n ≡ L [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n)
          ⊤]
have obtain : ∀ (L : ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
  (hL : Membership.mem (γ := Set ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) (⋂ (i : ℕ), S i) L),
  ∃ (L : ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))),
    ∀ (n : ℕ),
      f n ≡ L [SMOD
        HSMul.hSMul (β :=
          Submodule ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))
            ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
          (HPow.hPow (α := Ideal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K)))
            (IsLocalRing.maximalIdeal ↥(Valuation.integer (Γ₀ := ValueGroupWithZero K) (valuation K))) n)
          ⊤]
Tactic: obtain ⟨L, hL⟩ := (h 0).isCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed S hS
      (by simp [S]) h