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
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 l₂ : 𝓝 (HMul.hMul (α := hat K) 1 1)
Tactic: congr
          rfl
          rfl
          rw [← one_mul (1 : hat K)]
-- 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 conv : Tendsto (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 (HMul.hMul (α := hat K) 1 1))
Tactic: conv =>
          congr
          rfl
          rfl
          rw [← one_mul (1 : hat K)]
-- 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 (fun (p : hat K × hat K) ↦ p.1 * p.2⁻¹) (𝓝 (1, 1)) (𝓝 (HMul.hMul (α := hat K) 1 1))
have this : Tendsto.{u_1, u_1} (α := hat K) Inv.inv (𝓝 (1, 1).2) (𝓝 1)
Tactic: refine
          Tendsto.mul continuous_fst.continuousAt (Tendsto.comp ?_ continuous_snd.continuousAt)
-- 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.{u_1, u_1} (α := hat K) Inv.inv (𝓝 (1, 1).2) (𝓝 1)
have h₁ : Eq.{u_1 + 1} (α := hat K) 1 1⁻¹
Tactic: convert! (continuousAt_inv₀ (zero_ne_one.symm : 1 ≠ (0 : hat K))).tendsto
-- 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
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)
⊢ ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
have this : 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)
Tactic: have : Tendsto (fun p : hat K × hat K => p.1 * p.2⁻¹) ((𝓝 1) ×ˢ (𝓝 1)) (𝓝 1) := by
        rw [← nhds_prod_eq]
        conv =>
          congr
          rfl
          rfl
          rw [← one_mul (1 : hat K)]
        refine
          Tendsto.mul continuous_fst.continuousAt (Tendsto.comp ?_ continuous_snd.continuousAt)
        convert! (continuousAt_inv₀ (zero_ne_one.symm : 1 ≠ (0 : hat K))).tendsto
        exact inv_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
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)
this :
  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)
⊢ ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
have rcases : ∀ (U : Set (hat K)) (U_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) U)
  (hU : ∀ (x x' : hat K), x ∈ U → x' ∈ U → (x, x').1 * (x, x').2⁻¹ ∈ V),
  ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
Tactic: rcases tendsto_prod_self_iff.mp this V V_in with ⟨U, U_in, hU⟩
-- 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
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)
this :
  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)
U : Set (hat K)
U_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) U
hU : ∀ (x x' : hat K), x ∈ U → x' ∈ U → (x, x').1 * (x, x').2⁻¹ ∈ V
hatKstar : Set (hat K) := {0}ᶜ
⊢ ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
have this₁ : Membership.mem (γ := Filter (hat K)) (𝓝 1) hatKstar
Tactic: have : hatKstar ∈ 𝓝 (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm
-- 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
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)
⊢ ∃ (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 this : ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
Tactic: have : ∃ V' ∈ 𝓝 (1 : hat K), (0 : hat K) ∉ V' ∧ ∀ (x) (_ : x ∈ V') (y) (_ : y ∈ V'),
      x * y⁻¹ ∈ V := by
      have : Tendsto (fun p : hat K × hat K => p.1 * p.2⁻¹) ((𝓝 1) ×ˢ (𝓝 1)) (𝓝 1) := by
        rw [← nhds_prod_eq]
        conv =>
          congr
          rfl
          rfl
          rw [← one_mul (1 : hat K)]
        refine
          Tendsto.mul continuous_fst.continuousAt (Tendsto.comp ?_ continuous_snd.continuousAt)
        convert! (continuousAt_inv₀ (zero_ne_one.symm : 1 ≠ (0 : hat K))).tendsto
        exact inv_one.symm
      rcases tendsto_prod_self_iff.mp this V V_in with ⟨U, U_in, hU⟩
      let hatKstar := ({0}ᶜ : Set <| hat K)
      have : hatKstar ∈ 𝓝 (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm
      exact ⟨U ∩ hatKstar, Filter.inter_mem U_in this,
        ⟨fun ⟨_, h'⟩ ↦ h' rfl, fun x ⟨hx, _⟩ y ⟨hy, _⟩ ↦ hU _ _  hx hy⟩⟩
-- 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)
this : ∃ (V' : Set (hat K)), Membership.mem (γ := Filter (hat K)) (𝓝 1) V' ∧ 0 ∉ V' ∧ ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
⊢ ∃ (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') (zeroV' : 0 ∉ V')
  (hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V),
  ∃ (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 this with ⟨V', V'_in, zeroV', hV'⟩
Cleared hypotheses: this
-- 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
⊢ (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
have l : Function.LeftInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
Tactic: have l : Function.LeftInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, mul_inv_cancel₀ h, mul_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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
⊢ (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
have r : Function.RightInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
Tactic: have r : Function.RightInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, inv_mul_cancel₀ h, mul_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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
r : Function.RightInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
⊢ (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
have c : Continuous (Y := hat K) fun (x : hat K) ↦ x * x₀⁻¹
Tactic: have c : Continuous fun x : hat K => x * x₀⁻¹ := by fun_prop
-- 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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
r : Function.RightInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
c : Continuous (Y := hat K) fun (x : hat K) ↦ x * x₀⁻¹
⊢ (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
have rw : (fun (x : hat K) ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀
Tactic: rw [image_eq_preimage_of_inverse l r]
-- 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
l : Function.LeftInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
r : Function.RightInverse (fun (x : hat K) ↦ x * x₀⁻¹) fun (x : hat K) ↦ x * x₀
c : Continuous (Y := hat K) fun (x : hat K) ↦ x * x₀⁻¹
⊢ (fun (x : hat K) ↦ x * x₀⁻¹) ⁻¹' V' ∈ 𝓝 x₀
have V'_in₁ : V' ∈ 𝓝 (x₀ * x₀⁻¹)
Tactic: rw [← mul_inv_cancel₀ h] at V'_in
Cleared hypotheses: V'_in, zeroV', hV', l, r, c
-- 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
⊢ ∃ (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 nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
Tactic: have nhds_right : (fun x => x * x₀) '' V' ∈ 𝓝 x₀ := by
      have l : Function.LeftInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, mul_inv_cancel₀ h, mul_one]
      have r : Function.RightInverse (fun x : hat K => x * x₀⁻¹) fun x : hat K => x * x₀ := by
        intro x
        simp only [mul_assoc, inv_mul_cancel₀ h, mul_one]
      have c : Continuous fun x : hat K => x * x₀⁻¹ := by fun_prop
      rw [image_eq_preimage_of_inverse l r]
      rw [← mul_inv_cancel₀ h] at V'_in
      exact c.continuousAt V'_in
-- 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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
⊢ ∃ (z₀ : K), ∃ y₀ ∈ V', (↑z₀ : hat K) = y₀ * x₀ ∧ z₀ ≠ 0
have rcases : ∀ (z₀ : K) (y₀ : hat K) (y₀_in : y₀ ∈ V') (H : y₀ * x₀ = (↑z₀ : hat K)),
  ∃ (z₀ : K), ∃ y₀ ∈ V', (↑z₀ : hat K) = y₀ * x₀ ∧ z₀ ≠ 0
Tactic: rcases Completion.denseRange_coe.mem_nhds nhds_right with ⟨z₀, y₀, y₀_in, H : y₀ * x₀ = z₀⟩
-- 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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
H : y₀ * x₀ = (↑z₀ : hat K)
⊢ ∃ (z₀ : K), ∃ y₀ ∈ V', (↑z₀ : hat K) = y₀ * x₀ ∧ z₀ ≠ 0
have refine : z₀ ≠ 0
Tactic: refine ⟨z₀, y₀, y₀_in, ⟨H.symm, ?_⟩⟩
-- 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
⊢ ∃ (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 this : ∃ (z₀ : K), ∃ y₀ ∈ V', (↑z₀ : hat K) = y₀ * x₀ ∧ z₀ ≠ 0
Tactic: have : ∃ z₀ : K, ∃ y₀ ∈ V', ↑z₀ = y₀ * x₀ ∧ z₀ ≠ 0 := by
      rcases Completion.denseRange_coe.mem_nhds nhds_right with ⟨z₀, y₀, y₀_in, H : y₀ * x₀ = z₀⟩
      refine ⟨z₀, y₀, y₀_in, ⟨H.symm, ?_⟩⟩
      rintro rfl
      exact mul_ne_zero (ne_of_mem_of_not_mem y₀_in zeroV') h 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
this : ∃ (z₀ : K), ∃ y₀ ∈ V', (↑z₀ : hat K) = y₀ * x₀ ∧ z₀ ≠ 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 inr : ∀ (z₀ : K) (y₀ : hat K) (y₀_in : y₀ ∈ V') (hz₀ : (↑z₀ : hat K) = y₀ * x₀) (z₀_ne : z₀ ≠ 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 this with ⟨z₀, y₀, y₀_in, hz₀, z₀_ne⟩
Cleared hypotheses: this
-- 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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : (↑z₀ : hat K) = y₀ * x₀
z₀_ne : z₀ ≠ 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 vz₀_ne : Ne.{u_2 + 1} (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
  ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
  (0 : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
Tactic: have vz₀_ne : v.restrict z₀ ≠ 0 := by rwa [Valuation.ne_zero_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
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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : (↑z₀ : hat K) = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne :
  Ne.{u_2 + 1} (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    (0 : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ 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)
have inr : Tendsto.{u_1, u_2} (β := ValueGroup₀ (F := Valuation K Γ₀) v)
  ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) :
    (a : K) → ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
  (Filter.comap Completion.coe' (𝓝 x₀))
  (𝓝 ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀) : Filter (ValueGroup₀ (F := Valuation K Γ₀) v))
Tactic: refine ⟨v.restrict z₀, ?_⟩
-- 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)
V' : Set (hat K)
V'_in : Membership.mem (γ := Filter (hat K)) (𝓝 1) V'
zeroV' : 0 ∉ V'
hV' : ∀ x ∈ V', ∀ y ∈ V', x * y⁻¹ ∈ V
nhds_right : (fun (x : hat K) ↦ x * x₀) '' V' ∈ 𝓝 x₀
z₀ : K
y₀ : hat K
y₀_in : y₀ ∈ V'
hz₀ : (↑z₀ : hat K) = y₀ * x₀
z₀_ne : z₀ ≠ 0
vz₀_ne :
  Ne.{u_2 + 1} (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀ : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    (0 : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
⊢ Tendsto.{u_1, u_2} (β := ValueGroup₀ (F := Valuation K Γ₀) v)
    ((⇑(Valuation.restrict v) : (a : K) → ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀)) :
      (a : K) → ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
    (Filter.comap Completion.coe' (𝓝 x₀))
    (𝓝 ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀) :
      Filter (ValueGroup₀ (F := Valuation K Γ₀) v))
have inr : ∀ᶠ (b : hat K) in 𝓝 x₀,
  ∀ (a : K),
    (↑a : hat K) = b →
      Eq.{u_2 + 1} (α := ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
        ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) a : ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
        ((Valuation.restrict v : (a : K) → ValueGroup₀ (↑v : K →*₀ Γ₀)) z₀ :
          ValueGroup₀ (F := K →*₀ Γ₀) (↑v : K →*₀ Γ₀))
Tactic: rw [WithZeroTopology.tendsto_of_ne_zero vz₀_ne, eventually_comap]