awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PartitionOfUnity (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : EMetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
δ : C(X, ℝ≥0)
hδ₀ : ∀ (x : X), LT.lt (α := ℝ) (0 : ℝ) ((ContinuousMap.coeNNRealReal.comp δ : (a : X) → ℝ) x : ℝ)
hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x (ENNReal.ofReal ((ContinuousMap.coeNNRealReal.comp δ : (a : X) → ℝ) x)) ⊆ U i
⊢ ∃ (δ : C(X, ℝ≥0)),
    (∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)) ∧
      ∀ (i : ι), ∀ x ∈ K i, closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i
have refine : ∀ (i : ι) (x : X) (hi : x ∈ K i), closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i
Tactic: refine ⟨δ, hδ₀, fun i x hi => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PartitionOfUnity (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : MetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
⊢ ∃ (δ : C(X, ℝ≥0)),
    (∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)) ∧
      ∀ (i : ι), ∀ x ∈ K i, closedBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ) ⊆ U i
have rcases : ∀ (δ : C(X, ℝ≥0)) (hδ0 : ∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0))
  (hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i),
  ∃ (δ : C(X, ℝ≥0)),
    (∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)) ∧
      ∀ (i : ι), ∀ x ∈ K i, closedBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ) ⊆ U i
Tactic: rcases Metric.exists_continuous_nnreal_forall_closedEBall_subset hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PartitionOfUnity (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : MetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
δ : C(X, ℝ≥0)
hδ0 : ∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)
hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i
⊢ ∃ (δ : C(X, ℝ≥0)),
    (∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)) ∧
      ∀ (i : ι), ∀ x ∈ K i, closedBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ) ⊆ U i
have refine : ∀ (i : ι) (x : X) (hx : x ∈ K i), closedBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ) ⊆ U i
Tactic: refine ⟨δ, hδ0, fun i x hx => ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.PartitionOfUnity (goal)
Goal:
ι : Type u_1
X : Type u_2
inst : MetricSpace X
K U : ι → Set X
hK : ∀ (i : ι), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (K i)
hU : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
hKU : ∀ (i : ι), K i ⊆ U i
hfin : LocallyFinite K
δ : C(X, ℝ≥0)
hδ0 : ∀ (x : X), LT.lt (α := ℝ≥0) (0 : ℝ≥0) ((δ : (a : X) → ℝ≥0) x : ℝ≥0)
hδ : ∀ (i : ι), ∀ x ∈ K i, closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i
i : ι
x : X
hx : x ∈ K i
⊢ closedBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ) ⊆ U i
have rw : closedEBall x (↑((δ : (a : X) → ℝ≥0) x) : ℝ≥0∞) ⊆ U i
Tactic: rw [← Metric.closedEBall_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ x ∈ eball x (ε / 2)
have h : 0 < ε / 2
Tactic: apply Metric.mem_eball_self
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ 0 < ε / 2
have h : ε ≠ 0 ∧ Ne (α := ℝ≥0∞) 2 ∞
Tactic: rw [ENNReal.div_pos_iff]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
⊢ let D := closure (eball x (ε / 2) ∩ C);
  Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ ediam D ≤ ε
have this : x ∈ eball x (ε / 2)
Tactic: have : x ∈ Metric.eball x (ε / 2) := by
    apply Metric.mem_eball_self
    rw [ENNReal.div_pos_iff]
    exact ⟨ne_of_gt ε_pos, by simp⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this : x ∈ eball x (ε / 2)
⊢ let D := closure (eball x (ε / 2) ∩ C);
  Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ ediam D ≤ ε
have this₁ : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
Tactic: have := hC.closure_nhds_inter x xC this Metric.isOpen_eball
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ let D := closure (eball x (ε / 2) ∩ C);
  Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ ediam D ≤ ε
have refine_1 : closure (eball x (ε / 2) ∩ C) ⊆ C
Tactic: refine ⟨this.1, this.2, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
refine_1 : closure (eball x (ε / 2) ∩ C) ⊆ C
⊢ let D := closure (eball x (ε / 2) ∩ C);
  Perfect D ∧ D.Nonempty ∧ D ⊆ C ∧ ediam D ≤ ε
have refine_2 : ediam (closure (eball x (ε / 2) ∩ C)) ≤ ε
Tactic: refine ⟨this.1, this.2, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ closure (eball x (ε / 2) ∩ C) ⊆ C
have refine_1 : eball x (ε / 2) ∩ C ⊆ C
Tactic: rw [IsClosed.closure_subset_iff hC.closed]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ediam (closure (eball x (ε / 2) ∩ C)) ≤ ε
have refine_2 : ediam (eball x (ε / 2) ∩ C) ≤ ε
Tactic: rw [Metric.ediam_closure]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ediam (eball x (ε / 2) ∩ C) ≤ ε
have refine_2 : ediam (eball x (ε / 2)) ≤ ε
Tactic: apply le_trans (Metric.ediam_mono inter_subset_left)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ediam (eball x (ε / 2)) ≤ ε
have h : ε = 2 * (ε / 2)
Tactic: convert! Metric.ediam_eball_le (x := x)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
⊢ ε = 2 * (ε / 2)
have h : Ne (α := ℝ≥0∞) 2 0
Tactic: rw [mul_comm, ENNReal.div_mul_cancel]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
ε_pos : 0 < ε
x : α
xC : x ∈ C
this_1 : x ∈ eball x (ε / 2)
this : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty
h : Ne (α := ℝ≥0∞) 2 0
⊢ ε = 2 * (ε / 2)
have h₁ : Ne (α := ℝ≥0∞) 2 ∞
Tactic: rw [mul_comm, ENNReal.div_mul_cancel]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
ε_pos : 0 < ε
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have rcases : ∀ (D₀ D₁ : Set α) (perf0 : Perfect D₀) (non0 : D₀.Nonempty) (sub0 : D₀ ⊆ C) (hdisj : Disjoint D₀ D₁)
  (perf1 : Perfect D₁) (non1 : D₁.Nonempty) (sub1 : D₁ ⊆ C),
  ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
Tactic: rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
ε_pos : 0 < ε
D₀ D₁ : Set α
perf0 : Perfect D₀
non0 : D₀.Nonempty
sub0 : D₀ ⊆ C
hdisj : Disjoint D₀ D₁
perf1 : Perfect D₁
non1 : D₁.Nonempty
sub1 : D₁ ⊆ C
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have obtain : ∀ (x₀ : α) (hx₀ : x₀ ∈ D₀),
  ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
Tactic: obtain ⟨x₀, hx₀⟩ := non0
Cleared hypotheses: non0
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
ε_pos : 0 < ε
D₀ D₁ : Set α
perf0 : Perfect D₀
sub0 : D₀ ⊆ C
hdisj : Disjoint D₀ D₁
perf1 : Perfect D₁
non1 : D₁.Nonempty
sub1 : D₁ ⊆ C
x₀ : α
hx₀ : x₀ ∈ D₀
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have obtain : ∀ (x₁ : α) (hx₁ : x₁ ∈ D₁),
  ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
Tactic: obtain ⟨x₁, hx₁⟩ := non1
Cleared hypotheses: non1
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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 : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
Tactic: rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
Tactic: rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
Tactic: rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
Tactic: rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
Tactic: rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
Tactic: rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have sub1' : closure (eball x₁ (ε / 2) ∩ D₁) ⊆ D₁
Tactic: rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
sub1' : closure (eball x₁ (ε / 2) ∩ D₁) ⊆ D₁
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have diam1 : ediam (closure (eball x₁ (ε / 2) ∩ D₁)) ≤ ε
Tactic: rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
sub1' : closure (eball x₁ (ε / 2) ∩ D₁) ⊆ D₁
diam1 : ediam (closure (eball x₁ (ε / 2) ∩ D₁)) ≤ ε
⊢ ∃ (C₀ : Set α) (C₁ : Set α),
    (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ ediam C₀ ≤ ε) ∧
      (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ ediam C₁ ≤ ε) ∧ Disjoint C₀ C₁
have x₂ : Disjoint (closure (eball x₀ (ε / 2) ∩ D₀)) (closure (eball x₁ (ε / 2) ∩ D₁))
Tactic: refine
    ⟨closure (Metric.eball x₀ (ε / 2) ∩ D₀), closure (Metric.eball x₁ (ε / 2) ∩ D₁),
      ⟨perf0', non0', sub0'.trans sub0, diam0⟩, ⟨perf1', non1', sub1'.trans sub1, diam1⟩, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
sub1' : closure (eball x₁ (ε / 2) ∩ D₁) ⊆ D₁
diam1 : ediam (closure (eball x₁ (ε / 2) ∩ D₁)) ≤ ε
⊢ Disjoint (closure (eball x₀ (ε / 2) ∩ D₀)) (closure (eball x₁ (ε / 2) ∩ D₁))
have apply : closure (eball x₀ (ε / 2) ∩ D₀) ≤ D₀
Tactic: apply Disjoint.mono _ _ hdisj
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
ε : ℝ≥0∞
hC : Perfect C
hnonempty : C.Nonempty
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
perf0' : Perfect (closure (eball x₀ (ε / 2) ∩ D₀))
non0' : (closure (eball x₀ (ε / 2) ∩ D₀)).Nonempty
sub0' : closure (eball x₀ (ε / 2) ∩ D₀) ⊆ D₀
diam0 : ediam (closure (eball x₀ (ε / 2) ∩ D₀)) ≤ ε
perf1' : Perfect (closure (eball x₁ (ε / 2) ∩ D₁))
non1' : (closure (eball x₁ (ε / 2) ∩ D₁)).Nonempty
sub1' : closure (eball x₁ (ε / 2) ∩ D₁) ⊆ D₁
diam1 : ediam (closure (eball x₁ (ε / 2) ∩ D₁)) ≤ ε
apply : closure (eball x₀ (ε / 2) ∩ D₀) ≤ D₀
⊢ Disjoint (closure (eball x₀ (ε / 2) ∩ D₀)) (closure (eball x₁ (ε / 2) ∩ D₁))
have apply₁ : closure (eball x₁ (ε / 2) ∩ D₁) ≤ D₁
Tactic: apply Disjoint.mono _ _ hdisj
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst_1 : CompleteSpace α
⊢ ∃ (f : (ℕ → Bool) → α), range f ⊆ C ∧ Continuous f ∧ Injective f
have obtain : ∀ (u : ℕ → ℝ≥0∞) (upos' : ∀ (n : ℕ), Membership.mem (γ := Set ℝ≥0∞) (Ioo 0 1) (u n)) (hu : Tendsto u atTop (nhds 0)),
  ∃ (f : (ℕ → Bool) → α), range f ⊆ C ∧ Continuous f ∧ Injective f
Tactic: obtain ⟨u, -, upos', hu⟩ := exists_seq_strictAnti_tendsto' (zero_lt_one' ℝ≥0∞)
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst_1 : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), Membership.mem (γ := Set ℝ≥0∞) (Ioo 0 1) (u n)
hu : Tendsto u atTop (nhds 0)
⊢ ∃ (f : (ℕ → Bool) → α), range f ⊆ C ∧ Continuous f ∧ Injective f
have upos : ∀ (n : ℕ), 0 < u n
Tactic: have upos := fun n => (upos' n).1
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst_1 : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), Membership.mem (γ := Set ℝ≥0∞) (Ioo 0 1) (u n)
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E : Set α // Perfect E ∧ E.Nonempty }
⊢ ∃ (f : (ℕ → Bool) → α), range f ⊆ C ∧ Continuous f ∧ Injective f
have choose : ∀ (C0 C1 : {C : Set α} → (hC : Perfect C) → (hnonempty : C.Nonempty) → {ε : ℝ≥0∞} → (hε : 0 < ε) → Set α)
  (h0 :
    ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
      Perfect (C0 (C := C) hC hnonempty hε) ∧
        (C0 (C := C) hC hnonempty hε).Nonempty ∧
          C0 (C := C) hC hnonempty hε ⊆ C ∧ ediam (C0 (C := C) hC hnonempty hε) ≤ ε)
  (h1 :
    ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
      Perfect (C1 (C := C) hC hnonempty hε) ∧
        (C1 (C := C) hC hnonempty hε).Nonempty ∧
          C1 (C := C) hC hnonempty hε ⊆ C ∧ ediam (C1 (C := C) hC hnonempty hε) ≤ ε)
  (hdisj :
    ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
      Disjoint (C0 (C := C) hC hnonempty hε) (C1 (C := C) hC hnonempty hε)),
  ∃ (f : (ℕ → Bool) → α), range f ⊆ C ∧ Continuous f ∧ Injective f
Tactic: choose C0 C1 h0 h1 hdisj using
    fun {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε) =>
    hC.small_diam_splitting hnonempty hε
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Perfect (goal)
Goal:
α : Type u_1
inst : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst_1 : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), Membership.mem (γ := Set ℝ≥0∞) (Ioo 0 1) (u n)
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E : Set α // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → (hC : Perfect C) → (hnonempty : C.Nonempty) → {ε : ℝ≥0∞} → (hε : 0 < ε) → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 (C := C) hC hnonempty hε) ∧
      (C0 (C := C) hC hnonempty hε).Nonempty ∧ C0 (C := C) hC hnonempty hε ⊆ C ∧ ediam (C0 (C := C) hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 (C := C) hC hnonempty hε) ∧
      (C1 (C := C) hC hnonempty hε).Nonempty ∧ C1 (C := C) hC hnonempty hε ⊆ C ∧ ediam (C1 (C := C) hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 (C := C) hC hnonempty hε) (C1 (C := C) hC hnonempty hε)
l : List Bool
ih : P
⊢ P
have property : Perfect (C0 (C := (↑ih : Set α)) sorry sorry (ε := u (l.length + 1)) sorry) ∧
  (C0 (C := (↑ih : Set α)) sorry sorry (ε := u (l.length + 1)) sorry).Nonempty
Tactic: use C0 ih.property.1 ih.property.2 (upos (l.length + 1))