Module: Mathlib.MeasureTheory.Measure.LevyProkhorovMetric

Goal state before:
pp: Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : 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 : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | 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),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
⊢ ↑(ofMeasure P).toMeasure B ≤ ↑(ofMeasure Q).toMeasure (thickening δ B) + ENNReal.ofReal δ

Tactic:
have subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B := by
    intro ω ω_in_U
    simp only [mem_iUnion, exists_prop] at ω_in_U
    obtain ⟨k, ⟨B_intersects, _⟩, ω_in_thEk⟩ := ω_in_U
    rw [mem_thickening_iff] at ω_in_thEk ⊢
    obtain ⟨w, w_in_Ek, w_near⟩ := ω_in_thEk
    obtain ⟨z, ⟨z_in_B, z_in_Ek⟩⟩ := B_intersects
    refine ⟨z, z_in_B, lt_of_le_of_lt (dist_triangle ω w z) ?_⟩
    apply lt_of_le_of_lt (add_le_add w_near.le <|
            (dist_le_diam_of_mem (Es_bdd k) w_in_Ek z_in_Ek).trans <| Es_diam k)
    linarith

Goal state after:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : 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 : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m ↦ Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J ↦ thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | 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),
    IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P
Q : ProbabilityMeasure Ω
hQ :
  ∀ i ⊆ Iio N,
    ↑P (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) < ↑Q (⋃ i_2 ∈ i, thickening (ε / 3) (Es i_2)) + ENNReal.ofReal (ε / 3)
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | (B ∩ Es i).Nonempty ∧ i ∈ Iio N}
B_subset : B ⊆ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ (⋃ j ∈ Iio N, Es j)ᶜ
subset_thickB : ⋃ i ∈ JB, thickening (ε / 3) (Es i) ⊆ thickening δ B
⊢ ↑(ofMeasure P).toMeasure B ≤ ↑(ofMeasure Q).toMeasure (thickening δ B) + ENNReal.ofReal δ