l3lab/ntp-mathlib huggingface dataset:
/-- The **Cantor-Bendixson Theorem**: Any closed subset of a second countable space
can be written as the union of a countable set and a perfect set.-/
theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
    (hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by
  obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
  let v := { U ∈ b | (U ∩ C).Countable }
  let V := ⋃ U ∈ v, U
  let D := C \ V
  have Vct : (V ∩ C).Countable := by
    simp only [iUnion_inter, mem_sep_iff]
    apply Countable.biUnion
    · exact Countable.mono (inter_subset_left _ _) bct
    · exact inter_subset_right _ _
  refine' ⟨V ∩ C, D, Vct, ⟨_, _⟩, _⟩
  · refine' hclosed.sdiff (isOpen_biUnion fun _ ↦ _)
    exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub
  · rw [preperfect_iff_nhds]
    intro x xD E xE
    have : ¬(E ∩ D).Countable := by
      intro h
      obtain ⟨U, hUb, xU, hU⟩ : ∃ U ∈ b, x ∈ U ∧ U ⊆ E :=
        (IsTopologicalBasis.mem_nhds_iff bbasis).mp xE
      have hU_cnt : (U ∩ C).Countable := by
        apply @Countable.mono _ _ (E ∩ D ∪ V ∩ C)
        · rintro y ⟨yU, yC⟩
          by_cases h : y ∈ V
          · exact mem_union_right _ (mem_inter h yC)
          · exact mem_union_left _ (mem_inter (hU yU) ⟨yC, h⟩)
        exact Countable.union h Vct
      have : U ∈ v := ⟨hUb, hU_cnt⟩
      apply xD.2
      exact mem_biUnion this xU
    by_contra! h
    exact absurd (Countable.mono h (Set.countable_singleton _)) this
  · 

Proof state:
case intro.intro.intro.refine'_3
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : Set.Countable b
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ Set.Countable (U ∩ C)}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : Set.Countable (V ∩ C)
⊢ C = V ∩ C ∪ D

Next tactic:
rw [inter_comm, inter_union_diff]
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  

Proof state:
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
⊢ ∃ D, Perfect D ∧ Set.Nonempty D ∧ D ⊆ C

Next tactic:
rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ ∃ D, Perfect D ∧ Set.Nonempty D ∧ D ⊆ C

Next tactic:
refine' ⟨D, ⟨Dperf, _⟩⟩
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ Set.Nonempty D ∧ D ⊆ C

Next tactic:
constructor
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · 

Proof state:
case intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ Set.Nonempty D

Next tactic:
rw [nonempty_iff_ne_empty]
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    

Proof state:
case intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ D ≠ ∅

Next tactic:
by_contra h
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    

Proof state:
case intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
h : D = ∅
⊢ False

Next tactic:
rw [h, union_empty] at VD
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    rw [h, union_empty] at VD
    

Proof state:
case intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V
h : D = ∅
⊢ False

Next tactic:
rw [VD] at hunc
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    rw [h, union_empty] at VD
    rw [VD] at hunc
    

Proof state:
case intro.intro.intro.intro.left
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
V : Set α
hunc : ¬Set.Countable V
D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V
h : D = ∅
⊢ False

Next tactic:
contradiction
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    rw [h, union_empty] at VD
    rw [VD] at hunc
    contradiction
  

Proof state:
case intro.intro.intro.intro.right
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ D ⊆ C

Next tactic:
rw [VD]
/-- Any uncountable closed set in a second countable space contains a nonempty perfect subset.-/
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by
  rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
  refine' ⟨D, ⟨Dperf, _⟩⟩
  constructor
  · rw [nonempty_iff_ne_empty]
    by_contra h
    rw [h, union_empty] at VD
    rw [VD] at hunc
    contradiction
  rw [VD]
  

Proof state:
case intro.intro.intro.intro.right
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
hunc : ¬Set.Countable C
V D : Set α
Vct : Set.Countable V
Dperf : Perfect D
VD : C = V ∪ D
⊢ D ⊆ V ∪ D

Next tactic:
exact subset_union_right _ _
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ let D := closure (EMetric.ball x (ε / 2) ∩ C);
  Perfect D ∧ Set.Nonempty D ∧ D ⊆ C ∧ EMetric.diam D ≤ ε

Next tactic:
have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ x ∈ EMetric.ball x (ε / 2)

Next tactic:
apply EMetric.mem_ball_self
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    

Proof state:
case h
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ 0 < ε / 2

Next tactic:
rw [ENNReal.div_pos_iff]
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    

Proof state:
case h
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ ε ≠ 0 ∧ 2 ≠ ⊤

Next tactic:
exact ⟨ne_of_gt ε_pos, by norm_num⟩
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by 

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ 2 ≠ ⊤

Next tactic:
norm_num
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this : x ∈ EMetric.ball x (ε / 2)
⊢ let D := closure (EMetric.ball x (ε / 2) ∩ C);
  Perfect D ∧ Set.Nonempty D ∧ D ⊆ C ∧ EMetric.diam D ≤ ε

Next tactic:
have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ let D := closure (EMetric.ball x (ε / 2) ∩ C);
  Perfect D ∧ Set.Nonempty D ∧ D ⊆ C ∧ EMetric.diam D ≤ ε

Next tactic:
refine' ⟨this.1, this.2, _, _⟩
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · 

Proof state:
case refine'_1
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ closure (EMetric.ball x (ε / 2) ∩ C) ⊆ C

Next tactic:
rw [IsClosed.closure_subset_iff hC.closed]
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    

Proof state:
case refine'_1
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ EMetric.ball x (ε / 2) ∩ C ⊆ C

Next tactic:
apply inter_subset_right
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  

Proof state:
case refine'_2
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ EMetric.diam (closure (EMetric.ball x (ε / 2) ∩ C)) ≤ ε

Next tactic:
rw [EMetric.diam_closure]
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  rw [EMetric.diam_closure]
  

Proof state:
case refine'_2
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ EMetric.diam (EMetric.ball x (ε / 2) ∩ C) ≤ ε

Next tactic:
apply le_trans (EMetric.diam_mono (inter_subset_left _ _))
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  rw [EMetric.diam_closure]
  apply le_trans (EMetric.diam_mono (inter_subset_left _ _))
  

Proof state:
case refine'_2
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ EMetric.diam (EMetric.ball x (ε / 2)) ≤ ε

Next tactic:
convert EMetric.diam_ball (x := x)
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  rw [EMetric.diam_closure]
  apply le_trans (EMetric.diam_mono (inter_subset_left _ _))
  convert EMetric.diam_ball (x := x)
  

Proof state:
case h.e'_4
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ ε = 2 * (ε / 2)

Next tactic:
rw [mul_comm, ENNReal.div_mul_cancel]
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  rw [EMetric.diam_closure]
  apply le_trans (EMetric.diam_mono (inter_subset_left _ _))
  convert EMetric.diam_ball (x := x)
  rw [mul_comm, ENNReal.div_mul_cancel] <;> 

Proof state:
case h.e'_4.h0
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ 2 ≠ 0

Next tactic:
norm_num
private theorem Perfect.small_diam_aux (ε_pos : 0 < ε) {x : α} (xC : x ∈ C) :
    let D := closure (EMetric.ball x (ε / 2) ∩ C)
    Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε := by
  have : x ∈ EMetric.ball x (ε / 2) := by
    apply EMetric.mem_ball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by norm_num⟩
  have := hC.closure_nhds_inter x xC this EMetric.isOpen_ball
  refine' ⟨this.1, this.2, _, _⟩
  · rw [IsClosed.closure_subset_iff hC.closed]
    apply inter_subset_right
  rw [EMetric.diam_closure]
  apply le_trans (EMetric.diam_mono (inter_subset_left _ _))
  convert EMetric.diam_ball (x := x)
  rw [mul_comm, ENNReal.div_mul_cancel] <;> 

Proof state:
case h.e'_4.hI
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
ε_pos : 0 < ε
x : α
xC : x ∈ C
this✝ : x ∈ EMetric.ball x (ε / 2)
this : Perfect (closure (EMetric.ball x (ε / 2) ∩ C)) ∧ Set.Nonempty (closure (EMetric.ball x (ε / 2) ∩ C))
⊢ 2 ≠ ⊤

Next tactic:
norm_num
/-- A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. -/
theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
    ∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by
  

Proof state:
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
hnonempty : Set.Nonempty C
ε_pos : 0 < ε
⊢ ∃ C₀ C₁,
    (Perfect C₀ ∧ Set.Nonempty C₀ ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ Set.Nonempty C₁ ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁

Next tactic:
rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
/-- A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. -/
theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
    ∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by
  rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
hnonempty : Set.Nonempty C
ε_pos : 0 < ε
D₀ D₁ : Set α
perf0 : Perfect D₀
non0 : Set.Nonempty D₀
sub0 : D₀ ⊆ C
hdisj : Disjoint D₀ D₁
perf1 : Perfect D₁
non1 : Set.Nonempty D₁
sub1 : D₁ ⊆ C
⊢ ∃ C₀ C₁,
    (Perfect C₀ ∧ Set.Nonempty C₀ ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ Set.Nonempty C₁ ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁

Next tactic:
cases' non0 with x₀ hx₀
/-- A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. -/
theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
    ∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by
  rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
  cases' non0 with x₀ hx₀
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
hnonempty : Set.Nonempty C
ε_pos : 0 < ε
D₀ D₁ : Set α
perf0 : Perfect D₀
sub0 : D₀ ⊆ C
hdisj : Disjoint D₀ D₁
perf1 : Perfect D₁
non1 : Set.Nonempty D₁
sub1 : D₁ ⊆ C
x₀ : α
hx₀ : x₀ ∈ D₀
⊢ ∃ C₀ C₁,
    (Perfect C₀ ∧ Set.Nonempty C₀ ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ Set.Nonempty C₁ ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁

Next tactic:
cases' non1 with x₁ hx₁
/-- A refinement of `Perfect.splitting` for metric spaces, where we also control
the diameter of the new perfect sets. -/
theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
    ∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
    (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by
  rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
  cases' non0 with x₀ hx₀
  cases' non1 with x₁ hx₁
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MetricSpace α
C : Set α
hC : Perfect C
ε : ℝ≥0∞
hnonempty : Set.Nonempty C
ε_pos : 0 < ε
D₀ D₁ : Set α
perf0 : Perfect D₀
sub0 : D₀ ⊆ C
hdisj : Disjoint D₀ D₁
perf1 : Perfect D₁
sub1 : D₁ ⊆ C
x₀ : α
hx₀ : x₀ ∈ D₀
x₁ : α
hx₁ : x₁ ∈ D₁
⊢ ∃ C₀ C₁,
    (Perfect C₀ ∧ Set.Nonempty C₀ ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ Set.Nonempty C₁ ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁

Next tactic:
rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩