awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
p : α × ℝ
hp : p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v
⊢ p.2 ∈ f p.1
have refine_3 : ∀ (h'p : p ∈ t), p.2 ∈ f p.1
Tactic: rcases Finset.mem_union.1 hp with (h'p | h'p)
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
p : α × ℝ
hp : p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v
refine_3 : ∀ (h'p : p ∈ t), p.2 ∈ f p.1
⊢ p.2 ∈ f p.1
have refine_3₁ : ∀ (h'p : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) p), p.2 ∈ f p.1
Tactic: rcases Finset.mem_union.1 hp with (h'p | h'p)
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
p : α × ℝ
hp : p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v
h'p : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) p
⊢ p.2 ∈ f p.1
have refine_3 : ∀ (p' : α) (p'v : p' ∈ v) (hp : (p', r p') ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v)
  (h'p : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) (p', r p')),
  (p', r p').2 ∈ f (p', r p').1
Tactic: rcases Finset.mem_image.1 h'p with ⟨p', p'v, rfl⟩
Cleared hypotheses: p, hp, h'p
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v, closedBall p.1 p.2) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
have h : s \ ⋃ p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v, closedBall p.1 p.2 = s' \ ⋃ x ∈ v, closedBall x (r x)
Tactic: convert! hμv using 2
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ s \ ⋃ p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v, closedBall p.1 p.2 = s' \ ⋃ x ∈ v, closedBall x (r x)
have h : s \
    ((⋃ x ∈ t, closedBall x.1 x.2) ∪
      ⋃ (x : α × ℝ),
        ⋃ (_ : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) x), closedBall x.1 x.2) =
  s' \ ⋃ x ∈ v, closedBall x (r x)
Tactic: rw [Finset.set_biUnion_union, ← diff_diff, Finset.set_biUnion_finset_image]
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
h :
  s \
      ((⋃ x ∈ t, closedBall x.1 x.2) ∪
        ⋃ (x : α × ℝ),
          ⋃ (_ : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) x),
            closedBall x.1 x.2) =
    s' \ ⋃ x ∈ v, closedBall x (r x)
⊢ s \ ⋃ p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v, closedBall p.1 p.2 = s' \ ⋃ x ∈ v, closedBall x (r x)
have h₁ : (s \ ⋃ x ∈ t, closedBall x.1 x.2) \
    ⋃ (x : α × ℝ),
      ⋃ (_ : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) x), closedBall x.1 x.2 =
  s' \ ⋃ x ∈ v, closedBall x (r x)
Tactic: rw [Finset.set_biUnion_union, ← diff_diff, Finset.set_biUnion_finset_image]
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
t : Finset (α × ℝ)
ht : P t
B : Set α := ⋃ p ∈ t, closedBall p.1 p.2
hB : B = ⋃ p ∈ t, closedBall p.1 p.2
B_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] B
s' : Set α := s \ B
r : (x : α) → ℝ
hr : ∀ x ∈ s', r x ∈ f x ∩ Ioo 0 1 ∧ Disjoint B (closedBall x (r x))
v : Finset α
vs' : (↑v : Set α) ⊆ s'
hμv :
  (μ : (a : Set α) → ℝ≥0∞) (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
hv : (↑v : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
h :
  s \
      ((⋃ x ∈ t, closedBall x.1 x.2) ∪
        ⋃ (x : α × ℝ),
          ⋃ (_ : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) x),
            closedBall x.1 x.2) =
    s' \ ⋃ x ∈ v, closedBall x (r x)
h₁ :
  (s \ ⋃ x ∈ t, closedBall x.1 x.2) \
      ⋃ (x : α × ℝ),
        ⋃ (_ : Membership.mem (γ := Finset (α × ℝ)) (Finset.image (fun (x : α) ↦ (x, r x)) v) x), closedBall x.1 x.2 =
    s' \ ⋃ x ∈ v, closedBall x (r x)
⊢ s \ ⋃ p ∈ t ∪ Finset.image (fun (x : α) ↦ (x, r x)) v, closedBall p.1 p.2 = s' \ ⋃ x ∈ v, closedBall x (r x)
have h₂ : (s \ ⋃ x ∈ t, closedBall x.1 x.2) \ ⋃ y ∈ v, closedBall (y, r y).1 (y, r y).2 = s' \ ⋃ x ∈ v, closedBall x (r x)
Tactic: rw [Finset.set_biUnion_union, ← diff_diff, Finset.set_biUnion_finset_image]
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
this :
  ∀ (t : Finset (α × ℝ)),
    P t →
      ∃ (u : Finset (α × ℝ)),
        t ⊆ u ∧
          P u ∧
            (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤
              HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
                ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
⊢ ∃ (t : Set (α × ℝ)),
    t.Countable ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
have choose! : ∀ (F : (t : Finset (α × ℝ)) → Finset (α × ℝ))
  (hF :
    ∀ (t : Finset (α × ℝ)),
      P t →
        t ⊆ F t ∧
          P (F t) ∧
            (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
              HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
                ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)),
  ∃ (t : Set (α × ℝ)),
    t.Countable ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
Tactic: choose! F hF using this
Cleared hypotheses: this
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
⊢ ∃ (t : Set (α × ℝ)),
    t.Countable ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
have u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Tactic: have u_succ : ∀ n : ℕ, u n.succ = F (u n) := fun n => by
    simp only [u, Function.comp_apply, Function.iterate_succ']
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
⊢ P (u 0)
have zero : ((↑∅ : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
  (∀ (a : α) (b : ℝ), Membership.mem (γ := Finset (α × ℝ)) ∅ (a, b) → a ∈ s) ∧
    ∀ (a : α) (b : ℝ), Membership.mem (γ := Finset (α × ℝ)) ∅ (a, b) → b ∈ f a
Tactic: simp only [P, u, Prod.forall, id, Function.iterate_zero]
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
n : ℕ
IH : P (u n)
⊢ P (u (n + 1))
have succ : P (F (u n))
Tactic: rw [u_succ]
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
n : ℕ
⊢ P (u n)
have zero : P (u 0)
Tactic: induction n with
    | zero =>
      simp only [P, u, Prod.forall, id, Function.iterate_zero]
      simp only [Finset.notMem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall₂_true_iff,
        and_self_iff, pairwiseDisjoint_empty]
    | succ n IH =>
      rw [u_succ]
      exact (hF (u n) IH).2.1
Cleared hypotheses: n
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
n : ℕ
zero : P (u 0)
⊢ P (u n)
have succ : ∀ (n : ℕ) (IH : P (u n)), P (u (n + 1))
Tactic: induction n with
    | zero =>
      simp only [P, u, Prod.forall, id, Function.iterate_zero]
      simp only [Finset.notMem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall₂_true_iff,
        and_self_iff, pairwiseDisjoint_empty]
    | succ n IH =>
      rw [u_succ]
      exact (hF (u n) IH).2.1
Cleared hypotheses: n
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
⊢ ∃ (t : Set (α × ℝ)),
    t.Countable ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
have Pu : ∀ (n : ℕ), P (u n)
Tactic: have Pu : ∀ n, P (u n) := by
    intro n
    induction n with
    | zero =>
      simp only [P, u, Prod.forall, id, Function.iterate_zero]
      simp only [Finset.notMem_empty, IsEmpty.forall_iff, Finset.coe_empty, forall₂_true_iff,
        and_self_iff, pairwiseDisjoint_empty]
    | succ n IH =>
      rw [u_succ]
      exact (hF (u n) IH).2.1
-- 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 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
F : (t : Finset (α × ℝ)) → Finset (α × ℝ)
hF :
  ∀ (t : Finset (α × ℝ)),
    P t →
      t ⊆ F t ∧
        P (F t) ∧
          (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ F t, closedBall p.1 p.2) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)
u : (n : ℕ) → Finset (α × ℝ) := fun (n : ℕ) ↦ F^[n] ∅
u_succ : ∀ (n : ℕ), u n.succ = F (u n)
Pu : ∀ (n : ℕ), P (u n)
⊢ ∃ (t : Set (α × ℝ)),
    t.Countable ∧
      (∀ p ∈ t, p.1 ∈ s) ∧
        (∀ p ∈ t, p.2 ∈ f p.1) ∧
          Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞) (0 : ℝ≥0∞) ∧
            t.PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2
have refine_1 : ∀ (p : α × ℝ), Membership.mem (γ := Set (α × ℝ)) (⋃ (n : ℕ), (↑(u n) : Set (α × ℝ))) p → p.1 ∈ s
Tactic: refine ⟨⋃ n, u n, countable_iUnion fun n => (u n).countable_toSet, ?_, ?_, ?_, ?_⟩