l3lab/ntp-mathlib huggingface dataset:
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      

Proof state:
case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
hF : NeBot F ∧ ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
⊢ NeBot (map (fun x => x⁻¹) F) ∧ ∀ (γ : Γ₀ˣ), ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
refine' ⟨hF.1.map _, _⟩
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      

Proof state:
case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
hF : NeBot F ∧ ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
⊢ ∀ (γ : Γ₀ˣ), ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
replace hF := hF.2
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      

Proof state:
case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
hF : ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
⊢ ∀ (γ : Γ₀ˣ), ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
intro γ
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      

Proof state:
case intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
hF : ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
γ : Γ₀ˣ
⊢ ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      

Proof state:
case intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
hF : ∀ (γ : Γ₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
clear hF
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      

Proof state:
case intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ ∃ M ∈ map (fun x => x⁻¹) F, ∀ x ∈ M, ∀ y ∈ M, v (y - x) < ↑γ

Next tactic:
use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      

Proof state:
case h
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ (fun x => x⁻¹) '' (M₀ ∩ M₁) ∈ map (fun x => x⁻¹) F ∧
    ∀ x ∈ (fun x => x⁻¹) '' (M₀ ∩ M₁), ∀ y ∈ (fun x => x⁻¹) '' (M₀ ∩ M₁), v (y - x) < ↑γ

Next tactic:
constructor
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · 

Proof state:
case h.left
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ (fun x => x⁻¹) '' (M₀ ∩ M₁) ∈ map (fun x => x⁻¹) F

Next tactic:
rw [mem_map]
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        

Proof state:
case h.left
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ (fun x => x⁻¹) ⁻¹' ((fun x => x⁻¹) '' (M₀ ∩ M₁)) ∈ F

Next tactic:
apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        

Proof state:
case h.left
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ M₀ ∩ M₁ ⊆ (fun x => x⁻¹) ⁻¹' ((fun x => x⁻¹) '' (M₀ ∩ M₁))

Next tactic:
exact subset_preimage_image _ _
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · 

Proof state:
case h.right
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ ∀ x ∈ (fun x => x⁻¹) '' (M₀ ∩ M₁), ∀ y ∈ (fun x => x⁻¹) '' (M₀ ∩ M₁), v (y - x) < ↑γ

Next tactic:
rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        

Proof state:
case h.right.intro.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
x : K
x_in₀ : x ∈ M₀
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
⊢ v ((fun x => x⁻¹) y - (fun x => x⁻¹) x) < ↑γ

Next tactic:
simp only [mem_setOf_eq]
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        simp only [mem_setOf_eq]
        

Proof state:
case h.right.intro.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
H₁ : ∀ x ∈ M₁, ∀ y ∈ M₁, v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
x : K
x_in₀ : x ∈ M₀
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
⊢ v (y⁻¹ - x⁻¹) < ↑γ

Next tactic:
specialize H₁ x x_in₁ y y_in₁
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        simp only [mem_setOf_eq]
        specialize H₁ x x_in₁ y y_in₁
        

Proof state:
case h.right.intro.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₀ : x ∈ M₀
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
H₁ : v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
⊢ v (y⁻¹ - x⁻¹) < ↑γ

Next tactic:
replace x_in₀ := H₀ x x_in₀
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        simp only [mem_setOf_eq]
        specialize H₁ x x_in₁ y y_in₁
        replace x_in₀ := H₀ x x_in₀
        

Proof state:
case h.right.intro.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
H₀ : ∀ x ∈ M₀, ↑γ₀ ≤ v x
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
H₁ : v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
x_in₀ : ↑γ₀ ≤ v x
⊢ v (y⁻¹ - x⁻¹) < ↑γ

Next tactic:
clear H₀
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        simp only [mem_setOf_eq]
        specialize H₁ x x_in₁ y y_in₁
        replace x_in₀ := H₀ x x_in₀
        clear H₀
        

Proof state:
case h.right.intro.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
H₁ : v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
x_in₀ : ↑γ₀ ≤ v x
⊢ v (y⁻¹ - x⁻¹) < ↑γ

Next tactic:
apply Valuation.inversion_estimate
/-- A valued field is completable. -/
instance (priority := 100) completable : CompletableTopField K :=
  { ValuedRing.separated with
    nice := by
      rintro F hF h0
      have : ∃ γ₀ : Γ₀ˣ, ∃ M ∈ F, ∀ x ∈ M, (γ₀ : Γ₀) ≤ v x := by
        rcases Filter.inf_eq_bot_iff.mp h0 with ⟨U, U_in, M, M_in, H⟩
        rcases Valued.mem_nhds_zero.mp U_in with ⟨γ₀, hU⟩
        exists γ₀, M, M_in
        intro x xM
        apply le_of_not_lt _
        intro hyp
        have : x ∈ U ∩ M := ⟨hU hyp, xM⟩
        rwa [H] at this
      rcases this with ⟨γ₀, M₀, M₀_in, H₀⟩
      rw [Valued.cauchy_iff] at hF ⊢
      refine' ⟨hF.1.map _, _⟩
      replace hF := hF.2
      intro γ
      rcases hF (min (γ * γ₀ * γ₀) γ₀) with ⟨M₁, M₁_in, H₁⟩
      clear hF
      use (fun x : K => x⁻¹) '' (M₀ ∩ M₁)
      constructor
      · rw [mem_map]
        apply mem_of_superset (Filter.inter_mem M₀_in M₁_in)
        exact subset_preimage_image _ _
      · rintro _ ⟨x, ⟨x_in₀, x_in₁⟩, rfl⟩ _ ⟨y, ⟨_, y_in₁⟩, rfl⟩
        simp only [mem_setOf_eq]
        specialize H₁ x x_in₁ y y_in₁
        replace x_in₀ := H₀ x x_in₀
        clear H₀
        apply Valuation.inversion_estimate
        · 

Proof state:
case h.right.intro.intro.intro.intro.intro.intro.y_ne
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
src✝ : SeparatedSpace K := ValuedRing.separated
F : Filter K
h0 : 𝓝 0 ⊓ F = ⊥
γ₀ : Γ₀ˣ
M₀ : Set K
M₀_in : M₀ ∈ F
γ : Γ₀ˣ
M₁ : Set K
M₁_in : M₁ ∈ F
x : K
x_in₁ : x ∈ M₁
y : K
left✝ : y ∈ M₀
y_in₁ : y ∈ M₁
H₁ : v (y - x) < ↑(min (γ * γ₀ * γ₀) γ₀)
x_in₀ : ↑γ₀ ≤ v x
⊢ x ≠ 0

Next tactic:
have : (v x : Γ₀) ≠ 0 := by
            intro h
            rw [h] at x_in₀
            simp at x_in₀