awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K Γ₀) v)ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : (MonoidWithZeroHom.ValueGroup₀ (F := Valuation K Γ₀) v)ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
left : y ∈ M₀
y_in₁ : y ∈ M₁
H₁ :
  (Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) (y - x) <
    (↑(min (γ * γ₀ * γ₀) γ₀) : MonoidWithZeroHom.ValueGroup₀ v)
x_in₀ :
  (↑γ₀ : MonoidWithZeroHom.ValueGroup₀ v) ≤
    (Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x
h :
  min (↑(γ * γ₀ * γ₀) : MonoidWithZeroHom.ValueGroup₀ v) (↑γ₀ : MonoidWithZeroHom.ValueGroup₀ v) ≤
    min
      ((↑γ : MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) *
        HMul.hMul (α := MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) (β :=
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
          ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
          ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)))
      ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x)
⊢ (↑(min (γ * γ₀ * γ₀) γ₀) : MonoidWithZeroHom.ValueGroup₀ v) ≤
    min
      ((↑γ : MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) *
        HMul.hMul (α := MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) (β :=
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
          ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
          ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
            MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)))
      ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x)
have h₁ : min (↑(γ * (γ₀ * γ₀)) : MonoidWithZeroHom.ValueGroup₀ v) (↑γ₀ : MonoidWithZeroHom.ValueGroup₀ v) ≤
  min
    ((↑γ : MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) *
      HMul.hMul (α := MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) (β :=
        MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
        ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
        ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x :
          MonoidWithZeroHom.ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)))
    ((Valuation.restrict v : (a : K) → MonoidWithZeroHom.ValueGroup₀ (↑v : K →*₀ Γ₀)) x)
Tactic: grw [Units.min_val, mul_assoc, Units.val_mul, Units.val_mul, x_in₀]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ IsClosedMap.{u_1, u_2} (X := K) (Y := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
have refine : ∀ (s : Set K),
  IsClosed[hv.toTopologicalSpace] s →
    s.Nonempty → IsClosed[topologicalSpace] ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) '' s)
Tactic: refine IsClosedMap.of_nonempty ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU : IsClosed[hv.toTopologicalSpace] U
hU' : U.Nonempty
⊢ IsClosed[topologicalSpace] ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) '' U)
have hU₁ : ∀ x ∉ U,
  ∃ (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ),
    U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - x)}
Tactic: simp only [← isOpen_compl_iff, isOpen_iff_mem_nhds, mem_compl_iff, mem_nhds, subset_compl_comm,
    compl_setOf, not_lt] at hU
Cleared hypotheses: hU
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU' : U.Nonempty
hU :
  ∀ x ∉ U,
    ∃ (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ),
      U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - x)}
⊢ IsClosed[topologicalSpace] ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) '' U)
have simp : 0 ∈ U ∨
  ∃ (γ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)),
    ¬γ = 0 ∧ U ⊆ (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici γ
Tactic: simp only [isClosed_iff, mem_image, map_eq_zero, exists_eq_right, ne_eq, image_subset_iff]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU' : U.Nonempty
hU :
  ∀ x ∉ U,
    ∃ (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ),
      U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - x)}
⊢ 0 ∈ U ∨
    ∃ (γ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)),
      ¬γ = 0 ∧ U ⊆ (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici γ
have refine : ∀ (h : 0 ∉ U),
  ∃ (γ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)),
    ¬γ = 0 ∧ U ⊆ (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici γ
Tactic: refine (em _).imp_right fun h ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU' : U.Nonempty
hU :
  ∀ x ∉ U,
    ∃ (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ),
      U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - x)}
h_1 : 0 ∉ U
γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ
h : U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - 0)}
⊢ ∃ (γ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)),
    ¬γ = 0 ∧ U ⊆ (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici γ
have h₁ : U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) a}
Tactic: simp only [sub_zero] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
U : Set K
hU' : U.Nonempty
hU :
  ∀ x ∉ U,
    ∃ (γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ),
      U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (a - x)}
h_1 : 0 ∉ U
γ : (ValueGroup₀ (F := Valuation K Γ₀) v)ˣ
h : U ⊆ {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) a}
⊢ ∃ (γ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)),
    ¬γ = 0 ∧ U ⊆ (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici γ
have refine : Subset.{u_1} (α := Set K)
  {a : K | (↑γ : ValueGroup₀ v) ≤ (Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) a}
  ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) ⁻¹' Ici (↑γ : ValueGroup₀ v))
Tactic: refine ⟨γ.1, γ.ne_zero, h.trans ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Continuous.{u_1, u_2} (X := hat K) extension
have refine : ∀ (b : hat K),
  ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 b)) (𝓝 c)
Tactic: refine Completion.isDenseInducing_coe.continuous_extend ?_
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
⊢ ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
have inl : ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
  Tendsto.{u_1, u_2} (α := K) (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
    (Filter.comap (β := hat K) Completion.coe' (𝓝 0)) (𝓝 c)
Tactic: rcases eq_or_ne x₀ 0 with (rfl | h)
Cleared hypotheses: x₀
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
inl :
  ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto.{u_1, u_2} (α := K) (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
      (Filter.comap (β := hat K) Completion.coe' (𝓝 0)) (𝓝 c)
⊢ ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
have inr : ∀ (h : x₀ ≠ 0),
  ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
Tactic: rcases eq_or_ne x₀ 0 with (rfl | h)
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto.{u_1, u_2} (α := K) (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀))
      (Filter.comap (β := hat K) Completion.coe' (𝓝 0)) (𝓝 c)
have inl : Tendsto.{u_1, u_2} (α := K) (β := ValueGroup₀ (F := Valuation K Γ₀) v)
  (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap (β := hat K) Completion.coe' (𝓝 0))
  (𝓝 0)
Tactic: refine ⟨0, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Tendsto.{u_1, u_2} (α := K) (β := ValueGroup₀ (F := Valuation K Γ₀) v)
    (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap (β := hat K) Completion.coe' (𝓝 0))
    (𝓝 0)
have inl : Tendsto.{u_1, u_2} (α := K) (β := ValueGroup₀ (F := Valuation K Γ₀) v)
  (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (𝓝 0) (𝓝 0)
Tactic: rw [← Completion.coe_zero, ← Completion.isDenseInducing_coe.isInducing.nhds_eq_comap]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
⊢ Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
have rw : Ne.{u_2 + 1} (α := Γ₀) 1 0
Tactic: rw [Valuation.map_one]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
⊢ Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
have this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
Tactic: have : (v (1 : K) : Γ₀) ≠ 0 := by
        rw [Valuation.map_one]
        exact zero_ne_one.symm
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
⊢ Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
have h₁ : Eq.{u_1 + 1} (α := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
  {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
Tactic: convert! Valued.locally_const this
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
⊢ Eq.{u_1 + 1} (α := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
    {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
have h₁ : ∀ (x : K),
  Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
x : K
⊢ Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
      x
have h₁ : Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
  Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
Tactic: rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
x : K
h₁ :
  Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
⊢ Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
      x
have h₂ : Membership.mem (α := Γ₀) (γ := Set Γ₀) {1} ((v : (a : K) → Γ₀) x : Γ₀) ↔
  Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
Tactic: rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
x : K
h₁ :
  Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
h₂ :
  Membership.mem (α := Γ₀) (γ := Set Γ₀) {1} ((v : (a : K) → Γ₀) x : Γ₀) ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
⊢ Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
      x
have h₃ : Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀) ↔
  Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
Tactic: rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
this : Ne.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀) (0 : Γ₀)
x : K
h₁ :
  Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
h₂ :
  Membership.mem (α := Γ₀) (γ := Set Γ₀) {1} ((v : (a : K) → Γ₀) x : Γ₀) ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
h₃ :
  Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀) ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) (1 : Γ₀)} x
⊢ Membership.mem (γ := Set K) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}) x ↔
    Membership.mem (γ := Set K) {y : K | Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) y : Γ₀) ((v : (a : K) → Γ₀) 1 : Γ₀)}
      x
have h₄ : Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀) ↔
  Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
Tactic: rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (hypothesis)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
⊢ ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
have preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
Tactic: have preimage_one : v ⁻¹' {(1 : Γ₀)} ∈ 𝓝 (1 : K) := by
      have : (v (1 : K) : Γ₀) ≠ 0 := by
        rw [Valuation.map_one]
        exact zero_ne_one.symm
      convert! Valued.locally_const this
      ext x
      rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
⊢ ∃ (V : Set (hat K)),
    Membership.mem (γ := Filter (hat K)) (𝓝 1) V ∧
      ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
have rwa : ∀
  (preimage_one :
    Membership.mem (γ := Filter K) (Filter.comap (β := hat K) Completion.coe' (𝓝 (↑1 : hat K)))
      (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})),
  ∃ (V : Set (hat K)),
    Membership.mem (γ := Filter (hat K)) (𝓝 1) V ∧
      ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
Tactic: rwa [Completion.isDenseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
Cleared hypotheses: preimage_one
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
rwa :
  ∀
    (preimage_one :
      Membership.mem (γ := Filter K) (Filter.comap (β := hat K) Completion.coe' (𝓝 (↑1 : hat K)))
        (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})),
    ∃ (V : Set (hat K)),
      Membership.mem (γ := Filter (hat K)) (𝓝 1) V ∧
        ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
⊢ ∃ (V : Set (hat K)),
    Membership.mem (γ := Filter (hat K)) (𝓝 1) V ∧
      ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
have rwa₁ : ∀
  (preimage_one :
    ∃ (t : Set (hat K)),
      Membership.mem (γ := Filter (hat K)) (𝓝 (↑1 : hat K)) t ∧
        Completion.coe' ⁻¹' t ⊆ preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1}),
  ∃ (V : Set (hat K)),
    Membership.mem (γ := Filter (hat K)) (𝓝 1) V ∧
      ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
Tactic: rwa [Completion.isDenseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
Cleared hypotheses: preimage_one
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
⊢ ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
have inr : ∀ (V : Set (hat K)) (V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V)
  (hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)),
  ∃ (c : ValueGroup₀ (F := Valuation K Γ₀) v),
    Tendsto (⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) (Filter.comap Completion.coe' (𝓝 x₀))
      (𝓝 c)
Tactic: obtain ⟨V, V_in, hV⟩ : ∃ V ∈ 𝓝 (1 : hat K), ∀ x : K, (x : hat K) ∈ V → (v x : Γ₀) = 1 := by
      rwa [Completion.isDenseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
V : Set (hat K)
V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V
hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
⊢ Tendsto (β := hat K) (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹)
    (SProd.sprod (α := Filter (hat K)) (β := Filter (hat K)) (𝓝 1) (𝓝 1)) (𝓝 1)
have rw : Tendsto (β := hat K) (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)
Tactic: rw [← nhds_prod_eq]
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
V : Set (hat K)
V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V
hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
| Tendsto (β := hat K) (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)
have f : fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
V : Set (hat K)
V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V
hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
f : fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹
| Tendsto (β := hat K) (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)
have l₁ : 𝓝 (1, 1)
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
V : Set (hat K)
V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V
hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
f : fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹
l₁ : 𝓝 (1, 1)
| Tendsto (β := hat K) (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 1)
have l₂ : 𝓝 1
Tactic: congr
-- Lean 4 proof step from Mathlib.Topology.Algebra.Valued.ValuedField (goal)
Goal:
K : Type u_1
inst : Field K
Γ₀ : Type u_2
inst_1 : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : Membership.mem (γ := Filter K) (𝓝 1) (preimage (β := Γ₀) (⇑v : (a : K) → Γ₀) {1})
V : Set (hat K)
V_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V
hV : ∀ (x : K), (↑x : hat K) ∈ V → Eq.{u_2 + 1} (α := Γ₀) ((v : (a : K) → Γ₀) x : Γ₀) (1 : Γ₀)
| 𝓝 1
have l₂ : 𝓝 (HMul.hMul (α := hat K) 1 1)
Tactic: rw [← one_mul (1 : hat K)]