l3lab/ntp-mathlib huggingface dataset:
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
⊢ ∃ v, ∃ (_ : v ⊆ s), hausdorffEdist u v ≤ δ

Next tactic:
let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
⊢ ∃ v, ∃ (_ : v ⊆ s), hausdorffEdist u v ≤ δ

Next tactic:
exists v, (fun x hx => hx.1 : v ⊆ s)
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
⊢ hausdorffEdist u v ≤ δ

Next tactic:
refine' hausdorffEdist_le_of_mem_edist _ _
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · 

Proof state:
case refine'_1
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
⊢ ∀ x ∈ u, ∃ y ∈ v, edist x y ≤ δ

Next tactic:
intro x hx
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        

Proof state:
case refine'_1
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
⊢ ∃ y ∈ v, edist x y ≤ δ

Next tactic:
have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by 

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
⊢ x ∈ univ

Next tactic:
simp
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        

Proof state:
case refine'_1
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
this : x ∈ ⋃ y ∈ s, ball y δ
⊢ ∃ y ∈ v, edist x y ≤ δ

Next tactic:
rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        

Proof state:
case refine'_1.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
this : x ∈ ⋃ y ∈ s, ball y δ
y : α
ys : y ∈ s
dy : x ∈ ball y δ
⊢ ∃ y ∈ v, edist x y ≤ δ

Next tactic:
have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by 

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
this : x ∈ ⋃ y ∈ s, ball y δ
y : α
ys : y ∈ s
dy : x ∈ ball y δ
⊢ edist y x < δ

Next tactic:
simp at dy
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by simp at dy; 

Proof state:
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
this : x ∈ ⋃ y ∈ s, ball y δ
y : α
ys : y ∈ s
dy : edist x y < δ
⊢ edist y x < δ

Next tactic:
rwa [edist_comm] at dy
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
        

Proof state:
case refine'_1.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
hx : x ∈ u
this✝ : x ∈ ⋃ y ∈ s, ball y δ
y : α
ys : y ∈ s
dy : x ∈ ball y δ
this : edist y x < δ
⊢ ∃ y ∈ v, edist x y ≤ δ

Next tactic:
exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
        exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
      · 

Proof state:
case refine'_2
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
⊢ ∀ x ∈ v, ∃ y ∈ u, edist x y ≤ δ

Next tactic:
rintro x ⟨_, ⟨y, yu, hy⟩⟩
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
        exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
      · rintro x ⟨_, ⟨y, yu, hy⟩⟩
        

Proof state:
case refine'_2.intro.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
u : Set α
v : Set α := {x | x ∈ s ∧ ∃ y ∈ u, edist x y < δ}
x : α
left✝ : x ∈ s
y : α
yu : y ∈ u
hy : edist x y < δ
⊢ ∃ y ∈ u, edist x y ≤ δ

Next tactic:
exact ⟨y, yu, le_of_lt hy⟩
/-- In a compact space, the type of closed subsets is compact. -/
instance Closeds.compactSpace [CompactSpace α] : CompactSpace (Closeds α) :=
  ⟨by
    /- by completeness, it suffices to show that it is totally bounded,
        i.e., for all ε>0, there is a finite set which is ε-dense.
        start from a set `s` which is ε-dense in α. Then the subsets of `s`
        are finitely many, and ε-dense for the Hausdorff distance. -/
    refine'
      isCompact_of_totallyBounded_isClosed (EMetric.totallyBounded_iff.2 fun ε εpos => _)
        isClosed_univ
    rcases exists_between εpos with ⟨δ, δpos, δlt⟩
    rcases EMetric.totallyBounded_iff.1
        (isCompact_iff_totallyBounded_isComplete.1 (@isCompact_univ α _ _)).1 δ δpos with
      ⟨s, fs, hs⟩
    -- s : Set α, fs : s.Finite, hs : univ ⊆ ⋃ (y : α) (H : y ∈ s), eball y δ
    -- we first show that any set is well approximated by a subset of `s`.
    have main : ∀ u : Set α, ∃ (v : _) (_ : v ⊆ s), hausdorffEdist u v ≤ δ := by
      intro u
      let v := { x : α | x ∈ s ∧ ∃ y ∈ u, edist x y < δ }
      exists v, (fun x hx => hx.1 : v ⊆ s)
      refine' hausdorffEdist_le_of_mem_edist _ _
      · intro x hx
        have : x ∈ ⋃ y ∈ s, ball y δ := hs (by simp)
        rcases mem_iUnion₂.1 this with ⟨y, ys, dy⟩
        have : edist y x < δ := by simp at dy; rwa [edist_comm] at dy
        exact ⟨y, ⟨ys, ⟨x, hx, this⟩⟩, le_of_lt dy⟩
      · rintro x ⟨_, ⟨y, yu, hy⟩⟩
        exact ⟨y, yu, le_of_lt hy⟩
    -- introduce the set F of all subsets of `s` (seen as members of `Closeds α`).
    

Proof state:
case intro.intro.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
s✝ : Set α
inst✝ : CompactSpace α
ε : ℝ≥0∞
εpos : ε > 0
δ : ℝ≥0∞
δpos : 0 < δ
δlt : δ < ε
s : Set α
fs : Set.Finite s
hs : univ ⊆ ⋃ y ∈ s, ball y δ
main : ∀ (u : Set α), ∃ v, ∃ (_ : v ⊆ s), hausdorffEdist u v ≤ δ
⊢ ∃ t, Set.Finite t ∧ univ ⊆ ⋃ y ∈ t, ball y ε

Next tactic:
let F := { f : Closeds α | (f : Set α) ⊆ s }