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

Proof state:
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)
Js_finite : {J | J ⊆ Iio N}.Finite
⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P,
    dist ((equiv (ProbabilityMeasure Ω)).symm x) ((equiv (ProbabilityMeasure Ω)).symm P) < ε

Tactic:
set Gs := (fun (J : Set ℕ) ↦ thickening (ε/3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}

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)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
⊢ ∀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
⊢ Continuous ⇑(equiv (ProbabilityMeasure Ω)).symm

Tactic:
rw [continuous_iff_continuousAt]

Result:
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
⊢ ∀ (x : ProbabilityMeasure Ω), ContinuousAt (⇑(equiv (ProbabilityMeasure Ω)).symm) x

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)
exhaust : Tendsto (⇑↑P ∘ fun n => ⋃ i, ⋃ (_ : i ≥ n), Es i) atTop (𝓝 0)
⊢ ∃ N, ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

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

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
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, ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

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

Tactic:
have third_ε_pos : 0 < ε / 3 := by linarith

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
third_ε_pos : 0 < 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 : 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}
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
⊢ ↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ i ∈ JB, thickening (ε / 3) (Es i)) +
      ↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ j ∈ Iio N, Es j)ᶜ ≤
    ↑((equiv (ProbabilityMeasure Ω)).symm Q) (thickening δ B) + ENNReal.ofReal δ

Tactic:
specialize hQ _ (show JB ⊆ Iio N from fun _ h ↦ h.2)

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
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
δ : ℝ
B : Set Ω
δ_gt : 2 * (ε / 3) < δ
x✝ : MeasurableSet B
JB : Set ℕ := {i | B ∩ Es i ≠ ∅ ∧ 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
hQ : ↑P (⋃ i ∈ JB, thickening (ε / 3) (Es i)) < ↑Q (⋃ i ∈ JB, thickening (ε / 3) (Es i)) + ENNReal.ofReal (ε / 3)
⊢ ↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ i ∈ JB, thickening (ε / 3) (Es i)) +
      ↑((equiv (ProbabilityMeasure Ω)).symm P) (⋃ j ∈ Iio N, Es j)ᶜ ≤
    ↑((equiv (ProbabilityMeasure Ω)).symm Q) (thickening δ B) + ENNReal.ofReal δ