awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : (x : α) → Set ℝ := fun (x : α) ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
r : α → ℝ
t : Set α
v_count : (graphOn r t).Countable
vs : ∀ p ∈ graphOn r t, p.1 ∈ s
vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1
μv : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞)
v_disj : (graphOn r t).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
have hinj : InjOn (β := α × ℝ) (fun (x : α) ↦ (x, r x)) t
Tactic: have hinj : InjOn (fun x ↦ (x, r x)) t := LeftInvOn.injOn (f₁' := Prod.fst) fun _ _ ↦ rfl
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : (x : α) → Set ℝ := fun (x : α) ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
r : α → ℝ
t : Set α
v_count : (graphOn r t).Countable
vs : ∀ p ∈ graphOn r t, p.1 ∈ s
vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1
μv : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞)
v_disj : (graphOn r t).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
hinj : InjOn (β := α × ℝ) (fun (x : α) ↦ (x, r x)) t
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
have vs₁ : ∀ ⦃x : α⦄, x ∈ t → x ∈ s
Tactic: simp only [graphOn, forall_mem_image, biUnion_image, hinj.pairwiseDisjoint_image] at *
Cleared hypotheses: vs, vg, μv, v_disj
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : (x : α) → Set ℝ := fun (x : α) ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
r : α → ℝ
t : Set α
v_count : (graphOn r t).Countable
vs : ∀ p ∈ graphOn r t, p.1 ∈ s
vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1
μv : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞)
v_disj : (graphOn r t).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
hinj : InjOn (β := α × ℝ) (fun (x : α) ↦ (x, r x)) t
vs₁ : ∀ ⦃x : α⦄, x ∈ t → x ∈ s
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
have vg₁ : ∀ ⦃x : α⦄, x ∈ t → r x ∈ g x
Tactic: simp only [graphOn, forall_mem_image, biUnion_image, hinj.pairwiseDisjoint_image] at *
Cleared hypotheses: vs, vg, μv, v_disj
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : (x : α) → Set ℝ := fun (x : α) ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
r : α → ℝ
t : Set α
v_count : (graphOn r t).Countable
vs : ∀ p ∈ graphOn r t, p.1 ∈ s
vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1
μv : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞)
v_disj : (graphOn r t).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
hinj : InjOn (β := α × ℝ) (fun (x : α) ↦ (x, r x)) t
vs₁ : ∀ ⦃x : α⦄, x ∈ t → x ∈ s
vg₁ : ∀ ⦃x : α⦄, x ∈ t → r x ∈ g x
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
have μv₁ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ y ∈ t, closedBall y (r y)) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: simp only [graphOn, forall_mem_image, biUnion_image, hinj.pairwiseDisjoint_image] at *
Cleared hypotheses: vs, vg, μv, v_disj
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
g : (x : α) → Set ℝ := fun (x : α) ↦ f x ∩ Ioo 0 (R x)
hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty
r : α → ℝ
t : Set α
v_count : (graphOn r t).Countable
vs : ∀ p ∈ graphOn r t, p.1 ∈ s
vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1
μv : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞)
v_disj : (graphOn r t).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
hinj : InjOn (β := α × ℝ) (fun (x : α) ↦ (x, r x)) t
vs₁ : ∀ ⦃x : α⦄, x ∈ t → x ∈ s
vg₁ : ∀ ⦃x : α⦄, x ∈ t → r x ∈ g x
μv₁ : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ y ∈ t, closedBall y (r y)) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
have v_disj₁ : t.PairwiseDisjoint (α := Set α) ((fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∘ fun (x : α) ↦ (x, r x))
Tactic: simp only [graphOn, forall_mem_image, biUnion_image, hinj.pairwiseDisjoint_image] at *
Cleared hypotheses: vs, vg, μv, v_disj
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have obtain : ∀ (u : Set α) (su : u ⊇ s) (u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u)
  (μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)),
  ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
Tactic: obtain ⟨u, su, u_open, μu⟩ : ∃ U, U ⊇ s ∧ IsOpen U ∧ μ U ≤ μ s + ε / 2 :=
    Set.exists_isOpen_le_add _ _
      (by
        simpa only [or_false, Ne, ENNReal.div_eq_zero_iff, ENNReal.ofNat_ne_top] using hε)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have this : ∀ x ∈ s, ∃ R > 0, ball x R ⊆ u
Tactic: have : ∀ x ∈ s, ∃ R > 0, ball x R ⊆ u := fun x hx =>
    Metric.mem_nhds_iff.1 (u_open.mem_nhds (su hx))
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
this : ∀ x ∈ s, ∃ R > 0, ball x R ⊆ u
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have choose! : ∀ (R : (x : α) → ℝ) (hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u),
  ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
Tactic: choose! R hR using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have obtain : ∀ (t0 : Set α) (r0 : α → ℝ) (t0_count : t0.Countable) (t0s : t0 ⊆ s) (hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x))
  (μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞))
  (t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)),
  ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
Tactic: obtain ⟨t0, r0, t0_count, t0s, hr0, μt0, t0_disj⟩ :
    ∃ (t0 : Set α) (r0 : α → ℝ), t0.Countable ∧ t0 ⊆ s ∧
      (∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)) ∧ μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0 ∧
        t0.PairwiseDisjoint fun x => closedBall x (r0 x) :=
    exists_disjoint_closedBall_covering_ae μ f s hf R fun x hx => (hR x hx).1
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have s's : s' ⊆ s
Tactic: have s's : s' ⊆ s := diff_subset
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have obtain : ∀ (N : ℕ) (τ : ℝ) (hτ : 1 < τ) (H : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)),
  ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
Tactic: obtain ⟨N, τ, hτ, H⟩ : ∃ N τ, 1 < τ ∧ IsEmpty (Besicovitch.SatelliteConfig α N τ) :=
    HasBesicovitchCovering.no_satelliteConfig
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have obtain : ∀ (v : Set α) (s'v : v ⊇ s') (v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v)
  (μv : (μ : (a : Set α) → ℝ≥0∞) v ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) (ε / 2 / (↑N : ℝ≥0∞))),
  ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
Tactic: obtain ⟨v, s'v, v_open, μv⟩ : ∃ v, v ⊇ s' ∧ IsOpen v ∧ μ v ≤ μ s' + ε / 2 / N :=
    Set.exists_isOpen_le_add _ _
      (by simp only [ne_eq, ENNReal.div_eq_zero_iff, hε, ENNReal.ofNat_ne_top, or_self,
          ENNReal.natCast_ne_top, not_false_eq_true])
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : (μ : (a : Set α) → ℝ≥0∞) v ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) (ε / 2 / (↑N : ℝ≥0∞))
x : α
hx : x ∈ s'
⊢ ∃ r1 ∈ f x ∩ Ioo 0 1, closedBall x r1 ⊆ v
have rcases : ∀ (r : ℝ) (rpos : r > 0) (hr : ball x r ⊆ v), ∃ r1 ∈ f x ∩ Ioo 0 1, closedBall x r1 ⊆ v
Tactic: rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with ⟨r, rpos, hr⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (goal)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : (μ : (a : Set α) → ℝ≥0∞) v ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) (ε / 2 / (↑N : ℝ≥0∞))
x : α
hx : x ∈ s'
r : ℝ
rpos : r > 0
hr : ball x r ⊆ v
⊢ ∃ r1 ∈ f x ∩ Ioo 0 1, closedBall x r1 ⊆ v
have rcases : ∀ (R' : ℝ) (hR' : R' ∈ f x ∩ Ioo 0 (min r 1)), ∃ r1 ∈ f x ∩ Ioo 0 1, closedBall x r1 ⊆ v
Tactic: rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with ⟨R', hR'⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Besicovitch (hypothesis)
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : SFinite μ
inst_6 : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] u
μu : (μ : (a : Set α) → ℝ≥0∞) u ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (ε / 2)
R : (x : α) → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t0, closedBall x (r0 x)) : ℝ≥0∞) (0 : ℝ≥0∞)
t0_disj : t0.PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] v
μv : (μ : (a : Set α) → ℝ≥0∞) v ≤ HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) (ε / 2 / (↑N : ℝ≥0∞))
⊢ ∃ (t : Set α) (r : α → ℝ),
    t.Countable ∧
      t ⊆ s ∧
        (∀ x ∈ t, r x ∈ f x) ∧
          s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧
            ∑' (x : (↑t : Type u_1)), (μ : (a : Set α) → ℝ≥0∞) (closedBall (↑x : α) (r (↑x : α))) ≤
              HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ε
have this : ∀ x ∈ s', ∃ r1 ∈ f x ∩ Ioo 0 1, closedBall x r1 ⊆ v
Tactic: have : ∀ x ∈ s', ∃ r1 ∈ f x ∩ Ioo (0 : ℝ) 1, closedBall x r1 ⊆ v := by
    intro x hx
    rcases Metric.mem_nhds_iff.1 (v_open.mem_nhds (s'v hx)) with ⟨r, rpos, hr⟩
    rcases hf x (s's hx) (min r 1) (lt_min rpos zero_lt_one) with ⟨R', hR'⟩
    exact
      ⟨R', ⟨hR'.1, hR'.2.1, hR'.2.2.trans_le (min_le_right _ _)⟩,
        Subset.trans (closedBall_subset_ball (hR'.2.2.trans_le (min_le_left _ _))) hr⟩