ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.isAddHaarMeasure_of_isCompact_nonempty_interior
@[to_additive
"If a left-invariant measure gives positive mass to some compact set with nonempty interior, then
it is an additive Haar measure."]
theorem isHaarMeasure_of_isCompact_nonempty_interior [TopologicalGroup G] [BorelSpace G]
    (μ : Measure G) [IsMulLeftInvariant μ] (K : Set G) (hK : IsCompact K)
    (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ∞) : IsHaarMeasure μ 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddLeftInvariant] (K : Set G) (hK : IsCompact K) (h'K : (interior K).Nonempty) (h : μ K ≠ 0) (h' : μ K ≠ ⊤)
Goal: μ.IsAddHaarMeasure
-- MeasureTheory.Measure.isHaarMeasure_map
@[to_additive
"The image of an additive Haar measure under a continuous surjective proper additive group
homomorphism is again an additive Haar measure. See also `AddEquiv.isAddHaarMeasure_map`."]
theorem isHaarMeasure_map [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [T2Space H] [TopologicalGroup H]
    (f : G →* H) (hf : Continuous f) (h_surj : Surjective f)
    (h_prop : Tendsto f (cocompact G) (cocompact H)) : IsHaarMeasure (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_4} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [T2Space H] [TopologicalGroup H] (f : G →* H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) (h_prop : Filter.Tendsto (⇑f) (Filter.cocompact G) (Filter.cocompact H))
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsHaarMeasure
-- MeasureTheory.Measure.isAddHaarMeasure_map
@[to_additive
"The image of an additive Haar measure under a continuous surjective proper additive group
homomorphism is again an additive Haar measure. See also `AddEquiv.isAddHaarMeasure_map`."]
theorem isHaarMeasure_map [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [T2Space H] [TopologicalGroup H]
    (f : G →* H) (hf : Continuous f) (h_surj : Surjective f)
    (h_prop : Tendsto f (cocompact G) (cocompact H)) : IsHaarMeasure (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_4} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [T2Space H] [TopologicalAddGroup H] (f : G →+ H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f) (h_prop : Filter.Tendsto (⇑f) (Filter.cocompact G) (Filter.cocompact H))
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsAddHaarMeasure
-- MeasureTheory.Measure.isHaarMeasure_map_of_isFiniteMeasure
@[to_additive
"The image of a finite additive Haar measure under a continuous surjective additive group
homomorphism is again an additive Haar measure. See also `isAddHaarMeasure_map`."]
theorem isHaarMeasure_map_of_isFiniteMeasure
    [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [IsFiniteMeasure μ]
    (f : G →* H) (hf : Continuous f) (h_surj : Surjective f) :
    IsHaarMeasure (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_4} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [MeasureTheory.IsFiniteMeasure μ] (f : G →* H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f)
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsHaarMeasure
-- MeasureTheory.Measure.isAddHaarMeasure_map_of_isFiniteMeasure
@[to_additive
"The image of a finite additive Haar measure under a continuous surjective additive group
homomorphism is again an additive Haar measure. See also `isAddHaarMeasure_map`."]
theorem isHaarMeasure_map_of_isFiniteMeasure
    [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [IsFiniteMeasure μ]
    (f : G →* H) (hf : Continuous f) (h_surj : Surjective f) :
    IsHaarMeasure (Measure.map f μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_4} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalAddGroup H] [MeasureTheory.IsFiniteMeasure μ] (f : G →+ H) (hf : Continuous ⇑f) (h_surj : Function.Surjective ⇑f)
Goal: (MeasureTheory.Measure.map (⇑f) μ).IsAddHaarMeasure
-- MulEquiv.isHaarMeasure_map
@[to_additive "A convenience wrapper for `MeasureTheory.Measure.isAddHaarMeasure_map`."]
nonrec theorem _root_.MulEquiv.isHaarMeasure_map [BorelSpace G] [TopologicalGroup G] {H : Type*}
    [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H]
    [TopologicalGroup H] (e : G ≃* H) (he : Continuous e) (hesymm : Continuous e.symm) :
    IsHaarMeasure (Measure.map e μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [Group G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsHaarMeasure] [BorelSpace G] [TopologicalGroup G] {H : Type u_4} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] (e : G ≃* H) (he : Continuous ⇑e) (hesymm : Continuous ⇑e.symm)
Goal: (MeasureTheory.Measure.map (⇑e) μ).IsHaarMeasure
-- AddEquiv.isAddHaarMeasure_map
@[to_additive "A convenience wrapper for `MeasureTheory.Measure.isAddHaarMeasure_map`."]
nonrec theorem _root_.MulEquiv.isHaarMeasure_map [BorelSpace G] [TopologicalGroup G] {H : Type*}
    [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H]
    [TopologicalGroup H] (e : G ≃* H) (he : Continuous e) (hesymm : Continuous e.symm) :
    IsHaarMeasure (Measure.map e μ) 
Arguments: {G : Type u_2} [MeasurableSpace G] [AddGroup G] [TopologicalSpace G] (μ : MeasureTheory.Measure G) [μ.IsAddHaarMeasure] [BorelSpace G] [TopologicalAddGroup G] {H : Type u_4} [AddGroup H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalAddGroup H] (e : G ≃+ H) (he : Continuous ⇑e) (hesymm : Continuous ⇑e.symm)
Goal: (MeasureTheory.Measure.map (⇑e) μ).IsAddHaarMeasure
-- ContinuousLinearEquiv.isAddHaarMeasure_map
theorem _root_.ContinuousLinearEquiv.isAddHaarMeasure_map
    {E F R S : Type*} [Semiring R] [Semiring S]
    [AddCommGroup E] [Module R E] [AddCommGroup F] [Module S F]
    [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F]
    [TopologicalAddGroup F]
    {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
    [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F]
    (L : E ≃SL[σ] F) (μ : Measure E) [IsAddHaarMeasure μ] :
    IsAddHaarMeasure (μ.map L) 
Arguments: {E : Type u_4} {F : Type u_5} {R : Type u_6} {S : Type u_7} [Semiring R] [Semiring S] [AddCommGroup E] [Module R E] [AddCommGroup F] [Module S F] [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F] [TopologicalAddGroup F] {σ : R →+* S} {σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] (L : E ≃SL[σ] F) (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure]
Goal: (MeasureTheory.Measure.map (⇑L) μ).IsAddHaarMeasure
-- AnalyticAt.aeval_polynomial
theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
    AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {A : Type u_3} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CommSemiring A] {z : E} [NormedRing B] [NormedAlgebra 𝕜 B] [Algebra A B] {f : E → B} (hf : AnalyticAt 𝕜 f z) (p : Polynomial A)
Goal: AnalyticAt 𝕜 (fun x => (Polynomial.aeval (f x)) p) z
-- AnalyticOn.aeval_polynomial
theorem AnalyticOn.aeval_polynomial (hf : AnalyticOn 𝕜 f s) (p : A[X]) :
    AnalyticOn 𝕜 (fun x ↦ aeval (f x) p) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {A : Type u_3} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CommSemiring A] {s : Set E} [NormedRing B] [NormedAlgebra 𝕜 B] [Algebra A B] {f : E → B} (hf : AnalyticOn 𝕜 f s) (p : Polynomial A)
Goal: AnalyticOn 𝕜 (fun x => (Polynomial.aeval (f x)) p) s
-- AnalyticOn.eval_polynomial
theorem AnalyticOn.eval_polynomial {A} [NormedCommRing A] [NormedAlgebra 𝕜 A] (p : A[X]) :
    AnalyticOn 𝕜 (eval · p) Set.univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {A : Type u_5} [NormedCommRing A] [NormedAlgebra 𝕜 A] (p : Polynomial A)
Goal: AnalyticOn 𝕜 (fun x => Polynomial.eval x p) Set.univ
-- AnalyticAt.aeval_mvPolynomial
theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (p : MvPolynomial σ A) :
    AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {A : Type u_3} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CommSemiring A] {z : E} [NormedCommRing B] [NormedAlgebra 𝕜 B] [Algebra A B] {σ : Type u_5} {f : E → σ → B} (hf : ∀ (i : σ), AnalyticAt 𝕜 (fun x => f x i) z) (p : MvPolynomial σ A)
Goal: AnalyticAt 𝕜 (fun x => (MvPolynomial.aeval (f x)) p) z
-- AnalyticOn.aeval_mvPolynomial
theorem AnalyticOn.aeval_mvPolynomial (hf : ∀ i, AnalyticOn 𝕜 (f · i) s) (p : MvPolynomial σ A) :
    AnalyticOn 𝕜 (fun x ↦ aeval (f x) p) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {A : Type u_3} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CommSemiring A] {s : Set E} [NormedCommRing B] [NormedAlgebra 𝕜 B] [Algebra A B] {σ : Type u_5} {f : E → σ → B} (hf : ∀ (i : σ), AnalyticOn 𝕜 (fun x => f x i) s) (p : MvPolynomial σ A)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.aeval (f x)) p) s
-- AnalyticOn.eval_continuousLinearMap
theorem AnalyticOn.eval_continuousLinearMap (f : E →L[𝕜] σ → B) (p : MvPolynomial σ B) :
    AnalyticOn 𝕜 (fun x ↦ eval (f x) p) Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} (f : E →L[𝕜] σ → B) (p : MvPolynomial σ B)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.eval (f x)) p) Set.univ
-- AnalyticOn.eval_continuousLinearMap'
theorem AnalyticOn.eval_continuousLinearMap' (f : σ → E →L[𝕜] B) (p : MvPolynomial σ B) :
    AnalyticOn 𝕜 (fun x ↦ eval (f · x) p) Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} (f : σ → E →L[𝕜] B) (p : MvPolynomial σ B)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.eval fun x_1 => (f x_1) x) p) Set.univ
-- AnalyticOn.eval_linearMap
theorem AnalyticOn.eval_linearMap (f : E →ₗ[𝕜] σ → B) (p : MvPolynomial σ B) :
    AnalyticOn 𝕜 (fun x ↦ eval (f x) p) Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] σ → B) (p : MvPolynomial σ B)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.eval (f x)) p) Set.univ
-- AnalyticOn.eval_linearMap'
theorem AnalyticOn.eval_linearMap' (f : σ → E →ₗ[𝕜] B) (p : MvPolynomial σ B) :
    AnalyticOn 𝕜 (fun x ↦ eval (f · x) p) Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : σ → E →ₗ[𝕜] B) (p : MvPolynomial σ B)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.eval fun x_1 => (f x_1) x) p) Set.univ
-- AnalyticOn.eval_mvPolynomial
theorem AnalyticOn.eval_mvPolynomial [Fintype σ] (p : MvPolynomial σ 𝕜) :
    AnalyticOn 𝕜 (eval · p) Set.univ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {σ : Type u_5} [CompleteSpace 𝕜] [Fintype σ] (p : MvPolynomial σ 𝕜)
Goal: AnalyticOn 𝕜 (fun x => (MvPolynomial.eval x) p) Set.univ
-- MeasureTheory.meas_le_of_le_of_forall_le_meas_thickening_add
lemma meas_le_of_le_of_forall_le_meas_thickening_add {ε₁ ε₂ : ℝ≥0∞} (μ ν : Measure Ω)
    (h_le : ε₁ ≤ ε₂) {B : Set Ω} (hε₁ : μ B ≤ ν (thickening ε₁.toReal B) + ε₁) :
    μ B ≤ ν (thickening ε₂.toReal B) + ε₂ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] {ε₁ ε₂ : ℝ≥0∞} (μ ν : MeasureTheory.Measure Ω) (h_le : ε₁ ≤ ε₂) {B : Set Ω} (hε₁ : μ B ≤ ν (Metric.thickening ε₁.toReal B) + ε₁)
Goal: μ B ≤ ν (Metric.thickening ε₂.toReal B) + ε₂
-- MeasureTheory.left_measure_le_of_levyProkhorovEDist_lt
lemma left_measure_le_of_levyProkhorovEDist_lt {μ ν : Measure Ω} {c : ℝ≥0∞}
    (h : levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B) :
    μ B ≤ ν (thickening c.toReal B) + c 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] {μ ν : MeasureTheory.Measure Ω} {c : ℝ≥0∞} (h : MeasureTheory.levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B)
Goal: μ B ≤ ν (Metric.thickening c.toReal B) + c
-- MeasureTheory.right_measure_le_of_levyProkhorovEDist_lt
lemma right_measure_le_of_levyProkhorovEDist_lt {μ ν : Measure Ω} {c : ℝ≥0∞}
    (h : levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B) :
    ν B ≤ μ (thickening c.toReal B) + c 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] {μ ν : MeasureTheory.Measure Ω} {c : ℝ≥0∞} (h : MeasureTheory.levyProkhorovEDist μ ν < c) {B : Set Ω} (B_mble : MeasurableSet B)
Goal: ν B ≤ μ (Metric.thickening c.toReal B) + c
-- MeasureTheory.levyProkhorovEDist_le_of_forall_add_pos_le
lemma levyProkhorovEDist_le_of_forall_add_pos_le (μ ν : Measure Ω) (δ : ℝ≥0∞)
    (h : ∀ ε B, 0 < ε → ε < ∞ → MeasurableSet B →
      μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧
      ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε) :
    levyProkhorovEDist μ ν ≤ δ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω) (δ : ℝ≥0∞) (h :
  ∀ (ε : ℝ≥0∞) (B : Set Ω),
    0 < ε →
      ε < ⊤ →
        MeasurableSet B →
          μ B ≤ ν (Metric.thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (Metric.thickening (δ + ε).toReal B) + δ + ε)
Goal: MeasureTheory.levyProkhorovEDist μ ν ≤ δ
-- MeasureTheory.levyProkhorovEDist_le_of_forall
lemma levyProkhorovEDist_le_of_forall (μ ν : Measure Ω) (δ : ℝ≥0∞)
    (h : ∀ ε B, δ < ε → ε < ∞ → MeasurableSet B →
        μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε) :
    levyProkhorovEDist μ ν ≤ δ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω) (δ : ℝ≥0∞) (h :
  ∀ (ε : ℝ≥0∞) (B : Set Ω),
    δ < ε →
      ε < ⊤ → MeasurableSet B → μ B ≤ ν (Metric.thickening ε.toReal B) + ε ∧ ν B ≤ μ (Metric.thickening ε.toReal B) + ε)
Goal: MeasureTheory.levyProkhorovEDist μ ν ≤ δ
-- MeasureTheory.levyProkhorovEDist_le_max_measure_univ
lemma levyProkhorovEDist_le_max_measure_univ (μ ν : Measure Ω) :
    levyProkhorovEDist μ ν ≤ max (μ univ) (ν univ) 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovEDist μ ν ≤ max (μ Set.univ) (ν Set.univ)
-- MeasureTheory.levyProkhorovEDist_lt_top
lemma levyProkhorovEDist_lt_top (μ ν : Measure Ω) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    levyProkhorovEDist μ ν < ∞ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: MeasureTheory.levyProkhorovEDist μ ν < ⊤
-- MeasureTheory.levyProkhorovEDist_ne_top
lemma levyProkhorovEDist_ne_top (μ ν : Measure Ω) [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
    levyProkhorovEDist μ ν ≠ ∞ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
Goal: MeasureTheory.levyProkhorovEDist μ ν ≠ ⊤
-- MeasureTheory.levyProkhorovEDist_self
lemma levyProkhorovEDist_self (μ : Measure Ω) :
    levyProkhorovEDist μ μ = 0 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovEDist μ μ = 0
-- MeasureTheory.levyProkhorovEDist_comm
lemma levyProkhorovEDist_comm (μ ν : Measure Ω) :
    levyProkhorovEDist μ ν = levyProkhorovEDist ν μ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovEDist μ ν = MeasureTheory.levyProkhorovEDist ν μ
-- MeasureTheory.levyProkhorovEDist_triangle
lemma levyProkhorovEDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure Ω) :
    levyProkhorovEDist μ κ ≤ levyProkhorovEDist μ ν + levyProkhorovEDist ν κ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν κ : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovEDist μ κ ≤ MeasureTheory.levyProkhorovEDist μ ν + MeasureTheory.levyProkhorovEDist ν κ
-- MeasureTheory.levyProkhorovDist_self
lemma levyProkhorovDist_self (μ : Measure Ω) :
    levyProkhorovDist μ μ = 0 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovDist μ μ = 0
-- MeasureTheory.levyProkhorovDist_comm
lemma levyProkhorovDist_comm (μ ν : Measure Ω) :
    levyProkhorovDist μ ν = levyProkhorovDist ν μ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] (μ ν : MeasureTheory.Measure Ω)
Goal: MeasureTheory.levyProkhorovDist μ ν = MeasureTheory.levyProkhorovDist ν μ
-- MeasureTheory.levyProkhorovDist_triangle
lemma levyProkhorovDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure Ω)
    [IsFiniteMeasure μ] [IsFiniteMeasure ν] [IsFiniteMeasure κ] :
    levyProkhorovDist μ κ ≤ levyProkhorovDist μ ν + levyProkhorovDist ν κ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν κ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] [MeasureTheory.IsFiniteMeasure κ]
Goal: MeasureTheory.levyProkhorovDist μ κ ≤ MeasureTheory.levyProkhorovDist μ ν + MeasureTheory.levyProkhorovDist ν κ
-- MeasureTheory.LevyProkhorov.dist_def
lemma LevyProkhorov.dist_def (μ ν : LevyProkhorov (ProbabilityMeasure Ω)) :
    dist μ ν = levyProkhorovDist μ.toMeasure ν.toMeasure 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν : MeasureTheory.LevyProkhorov (MeasureTheory.ProbabilityMeasure Ω))
Goal: dist μ ν = MeasureTheory.levyProkhorovDist ↑μ ↑ν
-- MeasureTheory.levyProkhorovEDist_le_of_forall_le
lemma levyProkhorovEDist_le_of_forall_le
    (μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] (δ : ℝ≥0∞)
    (h : ∀ ε B, δ < ε → ε < ∞ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε) :
    levyProkhorovEDist μ ν ≤ δ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] (δ : ℝ≥0∞) (h : ∀ (ε : ℝ≥0∞) (B : Set Ω), δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (Metric.thickening ε.toReal B) + ε)
Goal: MeasureTheory.levyProkhorovEDist μ ν ≤ δ
-- MeasureTheory.levyProkhorovDist_le_of_forall_le
lemma levyProkhorovDist_le_of_forall_le
    (μ ν : Measure Ω) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] {δ : ℝ} (δ_nn : 0 ≤ δ)
    (h : ∀ ε B, δ < ε → MeasurableSet B → μ B ≤ ν (thickening ε B) + ENNReal.ofReal ε) :
    levyProkhorovDist μ ν ≤ δ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] {δ : ℝ} (δ_nn : 0 ≤ δ) (h : ∀ (ε : ℝ) (B : Set Ω), δ < ε → MeasurableSet B → μ B ≤ ν (Metric.thickening ε B) + ENNReal.ofReal ε)
Goal: MeasureTheory.levyProkhorovDist μ ν ≤ δ
-- MeasureTheory.BoundedContinuousFunction.integral_eq_integral_meas_le_of_hasFiniteIntegral
lemma BoundedContinuousFunction.integral_eq_integral_meas_le_of_hasFiniteIntegral
    {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α]
    (f : α →ᵇ ℝ) (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f) (hf : HasFiniteIntegral f μ) :
    ∫ ω, f ω ∂μ = ∫ t in Ioc 0 ‖f‖, ENNReal.toReal (μ {a : α | t ≤ f a}) 
Arguments: {α : Type u_3} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] ⇑f) (hf : MeasureTheory.HasFiniteIntegral (⇑f) μ)
Goal: ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Set.Ioc 0 ‖f‖, (μ {a | t ≤ f a}).toReal
-- MeasureTheory.BoundedContinuousFunction.integral_eq_integral_meas_le
lemma BoundedContinuousFunction.integral_eq_integral_meas_le
    {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α]
    (f : α →ᵇ ℝ) (μ : Measure α) [IsFiniteMeasure μ] (f_nn : 0 ≤ᵐ[μ] f) :
    ∫ ω, f ω ∂μ = ∫ t in Ioc 0 ‖f‖, ENNReal.toReal (μ {a : α | t ≤ f a}) 
Arguments: {α : Type u_3} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f_nn : 0 ≤ᵐ[μ] ⇑f)
Goal: ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Set.Ioc 0 ‖f‖, (μ {a | t ≤ f a}).toReal
-- MeasureTheory.BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt
lemma BoundedContinuousFunction.integral_le_of_levyProkhorovEDist_lt (μ ν : Measure Ω)
    [IsFiniteMeasure μ] [IsFiniteMeasure ν] {ε : ℝ} (ε_pos : 0 < ε)
    (hμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε) (f : Ω →ᵇ ℝ) (f_nn : 0 ≤ᵐ[μ] f) :
    ∫ ω, f ω ∂μ
      ≤ (∫ t in Ioc 0 ‖f‖, ENNReal.toReal (ν (thickening ε {a | t ≤ f a}))) + ε * ‖f‖ 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoMetricSpace Ω] [OpensMeasurableSpace Ω] (μ ν : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] {ε : ℝ} (ε_pos : 0 < ε) (hμν : MeasureTheory.levyProkhorovEDist μ ν < ENNReal.ofReal ε) (f : BoundedContinuousFunction Ω ℝ) (f_nn : 0 ≤ᵐ[μ] ⇑f)
Goal: ∫ (ω : Ω), f ω ∂μ ≤ (∫ (t : ℝ) in Set.Ioc 0 ‖f‖, (ν (Metric.thickening ε {a | t ≤ f a})).toReal) + ε * ‖f‖
-- MeasureTheory.tendsto_integral_meas_thickening_le
lemma tendsto_integral_meas_thickening_le (f : Ω →ᵇ ℝ)
    {A : Set ℝ} (A_finmeas : volume A ≠ ∞) (μ : ProbabilityMeasure Ω) :
    Tendsto (fun ε ↦ ∫ t in A, ENNReal.toReal (μ (thickening ε {a | t ≤ f a}))) (𝓝[>] (0 : ℝ))
      (𝓝 (∫ t in A, ENNReal.toReal (μ {a | t ≤ f a}))) 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoMetricSpace Ω] [OpensMeasurableSpace Ω] (f : BoundedContinuousFunction Ω ℝ) {A : Set ℝ} (A_finmeas : MeasureTheory.volume A ≠ ⊤) (μ : MeasureTheory.ProbabilityMeasure Ω)
Goal: Filter.Tendsto (fun ε => ∫ (t : ℝ) in A, (↑(μ (Metric.thickening ε {a | t ≤ f a}))).toReal) (𝓝[>] 0)
  (𝓝 (∫ (t : ℝ) in A, (↑(μ {a | t ≤ f a})).toReal))
-- MeasureTheory.LevyProkhorov.continuous_toProbabilityMeasure
lemma LevyProkhorov.continuous_toProbabilityMeasure :
    Continuous (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoMetricSpace Ω] [OpensMeasurableSpace Ω]
Goal: Continuous MeasureTheory.LevyProkhorov.toProbabilityMeasure
-- MeasureTheory.levyProkhorov_le_convergenceInDistribution
theorem levyProkhorov_le_convergenceInDistribution :
    TopologicalSpace.coinduced (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) inferInstance
      ≤ (inferInstance : TopologicalSpace (ProbabilityMeasure Ω)) 
Arguments: {Ω : Type u_2} [MeasurableSpace Ω] [PseudoMetricSpace Ω] [OpensMeasurableSpace Ω]
Goal: TopologicalSpace.coinduced MeasureTheory.LevyProkhorov.toProbabilityMeasure inferInstance ≤ inferInstance