awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
rw : ∀ (x : ProbabilityMeasure Ω), ContinuousAt ofMeasure x
⊢ Continuous[ProbabilityMeasure.instTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ofMeasure

Tactic:
rw [continuous_iff_continuousAt]

Premises used: continuous_iff_continuousAt
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
rw : ∀ ε > 0, ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε
⊢ ContinuousAt ofMeasure P

Tactic:
rw [continuousAt_iff']

Premises used: Metric.continuousAt_iff'
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
linarith : ε / 3 ≤ 0 → 3 / 3 = 1
linarith₁ : ε / 3 ≤ 0 → HMul.hMul (α := ℝ) 1 3 = 3
linarith₂ : ε / 3 ≤ 0 → GT.gt (α := ℝ) 3 0
linarith₃ : ε / 3 ≤ 0 → 0 - ε + (1 * ε - HMul.hMul (α := ℝ) 3 0) = 0
⊢ 0 < ε / 3

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
⊢ 0 < ε / 3

Tactic:
have third_ε_pos : 0 < ε / 3 := by linarith
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
⊢ 0 < ENNReal.ofReal (ε / 3)

Tactic:
have third_ε_pos' : 0 < ENNReal.ofReal (ε / 3) := ofReal_pos.mpr third_ε_pos
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
obtain :
  ∀ (Es : ℕ → Set Ω) (Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)) (Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n))
    (Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3) (Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ)
    (Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)),
    ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε

Tactic:
obtain ⟨Es, Es_mble, Es_bdd, Es_diam, Es_cover, Es_disjoint⟩ :=
    SeparableSpace.exists_measurable_partition_diam_le Ω third_ε_pos

Premises used: MeasureTheory.SeparableSpace.exists_measurable_partition_diam_le
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
⊢ Tendsto (α := ℕ) (β := ℝ≥0∞)
    (Function.comp (β := Set Ω) (⇑(↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) fun (n : ℕ) ↦ ⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i)
    atTop (𝓝 0)

Tactic:
have exhaust := @tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint Ω _ P.toMeasure _
                    Es (fun n ↦ (Es_mble n).nullMeasurableSet) Es_disjoint
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  Tendsto (α := ℕ) (β := ℝ≥0∞)
    (Function.comp (β := Set Ω) (⇑(↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) fun (n : ℕ) ↦ ⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i)
    atTop (𝓝 0)
⊢ ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)

Tactic:
simp only [tendsto_atTop_nhds, Function.comp_apply] at exhaust

Premises used: tendsto_atTop_nhds, Function.comp_apply
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
obtain :
  ∀ (N : ℕ)
    (hN :
      ∀ (n : ℕ),
        N ≤ n →
          Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
            (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)),
    ∃ (N : ℕ), ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
⊢ ∃ (N : ℕ), ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

Tactic:
obtain ⟨N, hN⟩ := exhaust (Iio (ENNReal.ofReal (ε / 3))) third_ε_pos' isOpen_Iio

Premises used: ENNReal.ofReal, isOpen_Iio, HDiv.hDiv, OfNat.ofNat, Set.Iio
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
N : ℕ
hN :
  ∀ (n : ℕ),
    N ≤ n →
      Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
        (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
⊢ ∃ (N : ℕ), ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

Tactic:
⟨N, ?_⟩

Premises used: Exists.intro
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
N : ℕ
hN :
  ∀ (n : ℕ),
    N ≤ n →
      Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
        (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
refine : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
⊢ ∃ (N : ℕ), ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

Tactic:
refine ⟨N, ?_⟩
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
N : ℕ
hN :
  ∀ (n : ℕ),
    N ≤ n →
      Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
        (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
⊢ Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), ⋃ (_ : N ≤ i), Es i) (⋃ (i : ℕ), ⋃ (_ : i < N), Es i)ᶜ

Tactic:
simpa only [mem_Iio, compl_Iio, mem_Ici] using
        (biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ Es_cover Es_disjoint (Iio N)).symm

Premises used: Set.mem_Iio, Set.mem_Ici, Set.biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ, Set.compl_Iio, Eq.symm, Set.Iio
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
N : ℕ
hN :
  ∀ (n : ℕ),
    N ≤ n →
      Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
        (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
⊢ Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), ⋃ (_ : N ≤ i), Es i) (⋃ (i : ℕ), ⋃ (_ : i < N), Es i)ᶜ

Tactic:
have rewr : ⋃ i, ⋃ (_ : N ≤ i), Es i = (⋃ i, ⋃ (_ : i < N), Es i)ᶜ := by
      simpa only [mem_Iio, compl_Iio, mem_Ici] using
        (biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ Es_cover Es_disjoint (Iio N)).symm
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
exhaust :
  ∀ (U : Set ℝ≥0∞),
    0 ∈ U →
      IsOpen U →
        ∃ (N : ℕ),
          ∀ (n : ℕ),
            N ≤ n →
              Membership.mem (α := ℝ≥0∞) U
                (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
N : ℕ
hN :
  ∀ (n : ℕ),
    N ≤ n →
      Membership.mem (α := ℝ≥0∞) (Iio (ENNReal.ofReal (ε / 3)))
        (((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ (i : ℕ), ⋃ (_ : i ≥ n), Es i) : ℝ≥0∞)
rewr : Eq.{u_1 + 1} (α := Set Ω) (⋃ (i : ℕ), ⋃ (_ : N ≤ i), Es i) (⋃ (i : ℕ), ⋃ (_ : i < N), Es i)ᶜ
⊢ ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

Tactic:
simpa only [mem_Iio, ← rewr, gt_iff_lt] using hN N le_rfl

Premises used: Set.mem_Iio, le_rfl, gt_iff_lt
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
obtain :
  ∀ (N : ℕ) (hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)),
    ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε

Tactic:
obtain ⟨N, hN⟩ : ∃ N, P.toMeasure (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3) := by
    have exhaust := @tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint Ω _ P.toMeasure _
                    Es (fun n ↦ (Es_mble n).nullMeasurableSet) Es_disjoint
    simp only [tendsto_atTop_nhds, Function.comp_apply] at exhaust
    obtain ⟨N, hN⟩ := exhaust (Iio (ENNReal.ofReal (ε / 3))) third_ε_pos' isOpen_Iio
    refine ⟨N, ?_⟩
    have rewr : ⋃ i, ⋃ (_ : N ≤ i), Es i = (⋃ i, ⋃ (_ : i < N), Es i)ᶜ := by
      simpa only [mem_Iio, compl_Iio, mem_Ici] using
        (biUnion_compl_eq_of_pairwise_disjoint_of_iUnion_eq_univ Es_cover Es_disjoint (Iio N)).symm
    simpa only [mem_Iio, ← rewr, gt_iff_lt] using hN N le_rfl

Premises used: ENNReal.ofReal, HDiv.hDiv, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
⊢ Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}

Tactic:
have Js_finite : Set.Finite {J | J ⊆ Iio N} := Finite.finite_subsets <| finite_Iio N
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}
Gs : Set (Set Ω) := (fun (J : Set ℕ) ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J : Set ℕ | J ⊆ Iio N}
⊢ ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))

Tactic:
have Gs_open : ∀ (J : Set ℕ), IsOpen (thickening (ε / 3) (⋃ j ∈ J, Es j)) :=
    fun J ↦ isOpen_thickening
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}
Gs : Set (Set Ω) := (fun (J : Set ℕ) ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J : Set ℕ | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
⊢ ∀ (G : Set Ω) (G_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G),
    {Q : ProbabilityMeasure Ω |
        ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G <
          HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G : ℝ≥0∞) (ENNReal.ofReal (ε / 3))} ∈
      𝓝 P

Tactic:
have mem_nhds_P (G : Set Ω) (G_open : IsOpen G) :
      {Q | P.toMeasure G < Q.toMeasure G + ENNReal.ofReal (ε/3)} ∈ 𝓝 P :=
    P.toMeasure_add_pos_gt_mem_nhds G_open third_ε_pos'
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}
Gs : Set (Set Ω) := (fun (J : Set ℕ) ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J : Set ℕ | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω) (G_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G),
    {Q : ProbabilityMeasure Ω |
        ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G <
          HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G : ℝ≥0∞) (ENNReal.ofReal (ε / 3))} ∈
      𝓝 P
h :
  ∀ (Q : ProbabilityMeasure Ω)
    (hQ :
      Membership.mem (γ := Set (ProbabilityMeasure Ω))
        (⋂ i ∈ Js_finite.toFinset (s := {J : Set ℕ | J ⊆ Iio N}),
          {Q : ProbabilityMeasure Ω |
            ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (thickening (ε / 3) (⋃ j ∈ i, Es j)) <
              HAdd.hAdd (α := ℝ≥0∞)
                (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (thickening (ε / 3) (⋃ j ∈ i, Es j)) : ℝ≥0∞)
                (ENNReal.ofReal (ε / 3))})
        Q),
    dist (ofMeasure Q) (ofMeasure P) < ε
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist (ofMeasure x) (ofMeasure P) < ε

Tactic:
filter_upwards [(Finset.iInter_mem_sets Js_finite.toFinset).mpr <|
                    fun J _ ↦ mem_nhds_P _ (Gs_open J)] with Q hQ

Premises used: Iff.mpr, HDiv.hDiv, Metric.thickening, Set.Finite.toFinset, Finset.iInter_mem_sets, OfNat.ofNat, Set.iUnion
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}
Gs : Set (Set Ω) := (fun (J : Set ℕ) ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J : Set ℕ | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω) (G_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G),
    {Q : ProbabilityMeasure Ω |
        ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G <
          HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G : ℝ≥0∞) (ENNReal.ofReal (ε / 3))} ∈
      𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  Membership.mem (γ := Set (ProbabilityMeasure Ω))
    (⋂ i ∈ Js_finite.toFinset (s := {J : Set ℕ | J ⊆ Iio N}),
      {Q : ProbabilityMeasure Ω |
        ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (thickening (ε / 3) (⋃ j ∈ i, Es j)) <
          HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (thickening (ε / 3) (⋃ j ∈ i, Es j)) : ℝ≥0∞)
            (ENNReal.ofReal (ε / 3))})
    Q
⊢ ∀ (i : Set ℕ) (i_1 : i ⊆ Iio N),
    ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) <
      HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) : ℝ≥0∞)
        (ENNReal.ofReal (ε / 3))

Tactic:
simp only [Finite.mem_toFinset, mem_setOf_eq, thickening_iUnion, mem_iInter] at hQ

Premises used: Set.Finite.mem_toFinset, Set.mem_setOf_eq, Metric.thickening_iUnion, Set.mem_iInter
-- Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
Goal:
Ω : Type u_1
inst : PseudoMetricSpace Ω
inst_1 : MeasurableSpace Ω
inst_2 : OpensMeasurableSpace Ω
inst_3 : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : Eq.{u_1 + 1} (α := Set Ω) (⋃ (n : ℕ), Es n) univ
Es_disjoint : Pairwise fun (n m : ℕ) ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : Set.Finite (α := Set ℕ) {J : Set ℕ | J ⊆ Iio N}
Gs : Set (Set Ω) := (fun (J : Set ℕ) ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J : Set ℕ | J ⊆ Iio N}
Gs_open :
  ∀ (J : Set ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P :
  ∀ (G : Set Ω) (G_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G),
    {Q : ProbabilityMeasure Ω |
        ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G <
          HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) G : ℝ≥0∞) (ENNReal.ofReal (ε / 3))} ∈
      𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ (i : Set ℕ) (i_1 : i ⊆ Iio N),
    ((↑P : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) <
      HAdd.hAdd (α := ℝ≥0∞) (((↑Q : Measure Ω) : (a : Set Ω) → ℝ≥0∞) (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) : ℝ≥0∞)
        (ENNReal.ofReal (ε / 3))
linarith : ε ≤ 2 * (ε / 3) → 3 / 3 = 1
linarith₁ linarith₂ : ε ≤ 2 * (ε / 3) → HMul.hMul (α := ℝ) 1 3 = 3
linarith₃ : ε ≤ 2 * (ε / 3) → GT.gt (α := ℝ) 3 0
linarith₄ : ε ≤ 2 * (ε / 3) → 3 / 3 = 1
linarith₅ : ε ≤ 2 * (ε / 3) → HMul.hMul (α := ℝ) 1 3 = 3
linarith₆ : ε ≤ 2 * (ε / 3) → GT.gt (α := ℝ) 3 0
linarith₇ : ε ≤ 2 * (ε / 3) → 0 - ε + (3 * ε - HMul.hMul (α := ℝ) 1 2 * (1 * ε)) = 0
⊢ 2 * (ε / 3) < ε

Tactic:
linarith