-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
    Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm

Proof state:
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 (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P : ∀ (G : Set Ω), IsOpen 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 ≠ ∅ ∧ i ∈ Iio N}
ω : Ω
ω_in_B : ω ∈ B
⊢ ω ∈ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ ⋃ j ∈ Ici N, Es j

Tactic:
obtain ⟨i, hi⟩ := show ∃ n, ω ∈ Es n by simp only [← mem_iUnion, Es_cover, mem_univ]

Result:
case intro
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 (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P : ∀ (G : Set Ω), IsOpen 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 ≠ ∅ ∧ i ∈ Iio N}
ω : Ω
ω_in_B : ω ∈ B
i : ℕ
hi : ω ∈ Es i
⊢ ω ∈ (⋃ i ∈ JB, thickening (ε / 3) (Es i)) ∪ ⋃ j ∈ Ici N, Es j

Proof state:
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)
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm 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

Result:
case intro.intro.intro.intro.intro
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)
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε

Proof state:
case intro
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)
exhaust : ∀ (U : Set ℝ≥0∞), 0 ∈ U → IsOpen U → ∃ N, ∀ (n : ℕ), N ≤ n → ↑P (⋃ i, ⋃ (_ : i ≥ n), Es i) ∈ U
N : ℕ
hN : ∀ (n : ℕ), N ≤ n → ↑P (⋃ i, ⋃ (_ : i ≥ n), Es i) ∈ Iio (ENNReal.ofReal (ε / 3))
rewr : ⋃ i, ⋃ (_ : N ≤ i), Es i = (⋃ i, ⋃ (_ : i < N), Es i)ᶜ
⊢ ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

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

Result:
no goals

Proof state:
case intro.intro.intro.intro.intro
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)
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm 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

Result:
case intro.intro.intro.intro.intro.intro
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)
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε

Proof state:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
⊢ ∀ ε > 0,
    ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
      dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε

Tactic:
intro ε ε_pos

Result:
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
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε