ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- analyticAt_fst
theorem analyticAt_fst {p : E × F} : AnalyticAt 𝕜 (fun p : E × F ↦ p.fst) p 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F}
Goal: AnalyticAt 𝕜 (fun p => p.1) p
-- analyticAt_snd
theorem analyticAt_snd {p : E × F} : AnalyticAt 𝕜 (fun p : E × F ↦ p.snd) p 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {p : E × F}
Goal: AnalyticAt 𝕜 (fun p => p.2) p
-- analyticOn_fst
theorem analyticOn_fst {s : Set (E × F)} : AnalyticOn 𝕜 (fun p : E × F ↦ p.fst) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set (E × F)}
Goal: AnalyticOn 𝕜 (fun p => p.1) s
-- analyticOn_snd
theorem analyticOn_snd {s : Set (E × F)} : AnalyticOn 𝕜 (fun p : E × F ↦ p.snd) s 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set (E × F)}
Goal: AnalyticOn 𝕜 (fun p => p.2) s
-- MeasureTheory.Measure.withDensity_rnDeriv_eq
theorem withDensity_rnDeriv_eq (μ ν : Measure α) [HaveLebesgueDecomposition μ ν] (h : μ ≪ ν) :
    ν.withDensity (rnDeriv μ ν) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [μ.HaveLebesgueDecomposition ν] (h : μ ≪ ν)
Goal: ν.withDensity (μ.rnDeriv ν) = μ
-- MeasureTheory.Measure.absolutelyContinuous_iff_withDensity_rnDeriv_eq
theorem absolutelyContinuous_iff_withDensity_rnDeriv_eq
    [HaveLebesgueDecomposition μ ν] : μ ≪ ν ↔ ν.withDensity (rnDeriv μ ν) = μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν]
Goal: μ ≪ ν ↔ ν.withDensity (μ.rnDeriv ν) = μ
-- MeasureTheory.Measure.rnDeriv_pos
lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
    ∀ᵐ x ∂μ, 0 < μ.rnDeriv ν x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν)
Goal: ∀ᵐ (x : α) ∂μ, 0 < μ.rnDeriv ν x
-- MeasureTheory.Measure.rnDeriv_pos'
lemma rnDeriv_pos' [SigmaFinite μ] [SFinite ν] (hμν : μ ≪ ν) :
    ∀ᵐ x ∂μ, 0 < ν.rnDeriv μ x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SFinite ν] (hμν : μ ≪ ν)
Goal: ∀ᵐ (x : α) ∂μ, 0 < ν.rnDeriv μ x
-- MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_left
lemma rnDeriv_withDensity_withDensity_rnDeriv_left (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ≥0∞} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: ((ν.withDensity (μ.rnDeriv ν)).withDensity f).rnDeriv ν =ᵐ[ν] (μ.withDensity f).rnDeriv ν
-- MeasureTheory.Measure.rnDeriv_withDensity_withDensity_rnDeriv_right
lemma rnDeriv_withDensity_withDensity_rnDeriv_right (μ ν : Measure α) [SigmaFinite μ]
    [SigmaFinite ν] (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0)
    (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ≥0∞} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ⊤)
Goal: (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᵐ[ν] μ.rnDeriv (ν.withDensity f)
-- MeasureTheory.Measure.rnDeriv_withDensity_left_of_absolutelyContinuous
lemma rnDeriv_withDensity_left_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν) :
    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν)
Goal: (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x => f x * μ.rnDeriv ν x
-- MeasureTheory.Measure.rnDeriv_withDensity_left
lemma rnDeriv_withDensity_left {μ ν : Measure α} [SigmaFinite μ] [SigmaFinite ν]
    (hfν : AEMeasurable f ν) (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
    (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x ↦ f x * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ≥0∞} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hfν : AEMeasurable f ν) (hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤)
Goal: (μ.withDensity f).rnDeriv ν =ᵐ[ν] fun x => f x * μ.rnDeriv ν x
-- MeasureTheory.Measure.rnDeriv_withDensity_right_of_absolutelyContinuous
lemma rnDeriv_withDensity_right_of_absolutelyContinuous {ν : Measure α} [SigmaFinite μ]
    [SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν)
    (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν) (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ⊤)
Goal: μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x
-- MeasureTheory.Measure.rnDeriv_withDensity_right
lemma rnDeriv_withDensity_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
    (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ x ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ x ∂ν, f x ≠ ∞) :
    μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x ↦ (f x)⁻¹ * μ.rnDeriv ν x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ≥0∞} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hf : AEMeasurable f ν) (hf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0) (hf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ⊤)
Goal: μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x
-- MeasureTheory.Measure.rnDeriv_eq_zero_of_mutuallySingular
lemma rnDeriv_eq_zero_of_mutuallySingular [SigmaFinite μ] {ν' : Measure α}
    [SigmaFinite ν'] (h : μ ⟂ₘ ν) (hνν' : ν ≪ ν') :
    μ.rnDeriv ν' =ᵐ[ν] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {ν' : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite ν'] (h : μ ⟂ₘ ν) (hνν' : ν ≪ ν')
Goal: μ.rnDeriv ν' =ᵐ[ν] 0
-- MeasureTheory.Measure.rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular
lemma rnDeriv_add_right_of_absolutelyContinuous_of_mutuallySingular {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
    (hμν : μ ≪ ν) (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ν'] (hμν : μ ≪ ν) (hνν' : ν ⟂ₘ ν')
Goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν
-- MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular'
lemma rnDeriv_add_right_of_mutuallySingular' {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν']
    (hμν' : μ ⟂ₘ ν') (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ν'] (hμν' : μ ⟂ₘ ν') (hνν' : ν ⟂ₘ ν')
Goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν
-- MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular
lemma rnDeriv_add_right_of_mutuallySingular {ν' : Measure α}
    [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν'] (hνν' : ν ⟂ₘ ν') :
    μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν ν' : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ν'] (hνν' : ν ⟂ₘ ν')
Goal: μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν
-- MeasureTheory.Measure.rnDeriv_withDensity_rnDeriv
lemma rnDeriv_withDensity_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν)
Goal: μ.rnDeriv (μ.withDensity (ν.rnDeriv μ)) =ᵐ[μ] μ.rnDeriv ν
-- MeasureTheory.Measure.inv_rnDeriv_aux
lemma inv_rnDeriv_aux [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ) :
    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν) (hνμ : ν ≪ μ)
Goal: (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ
-- MeasureTheory.Measure.inv_rnDeriv
lemma inv_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν)
Goal: (μ.rnDeriv ν)⁻¹ =ᵐ[μ] ν.rnDeriv μ
-- MeasureTheory.Measure.inv_rnDeriv'
lemma inv_rnDeriv' [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν)
Goal: (ν.rnDeriv μ)⁻¹ =ᵐ[μ] μ.rnDeriv ν
-- MeasureTheory.Measure.setLIntegral_rnDeriv_le
lemma setLIntegral_rnDeriv_le (s : Set α) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν ≤ μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (s : Set α)
Goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν ≤ μ s
-- MeasureTheory.Measure.setLIntegral_rnDeriv'
lemma setLIntegral_rnDeriv' [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {s : Set α}
    (hs : MeasurableSet s) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν = μ s
-- MeasureTheory.Measure.setLIntegral_rnDeriv
lemma setLIntegral_rnDeriv [HaveLebesgueDecomposition μ ν] [SFinite ν]
    (hμν : μ ≪ ν) (s : Set α) :
    ∫⁻ x in s, μ.rnDeriv ν x ∂ν = μ s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] [MeasureTheory.SFinite ν] (hμν : μ ≪ ν) (s : Set α)
Goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν = μ s
-- MeasureTheory.Measure.lintegral_rnDeriv
lemma lintegral_rnDeriv [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
    ∫⁻ x, μ.rnDeriv ν x ∂ν = μ Set.univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν)
Goal: ∫⁻ (x : α), μ.rnDeriv ν x ∂ν = μ Set.univ
-- MeasureTheory.Measure.integrableOn_toReal_rnDeriv
lemma integrableOn_toReal_rnDeriv {s : Set α} (hμs : μ s ≠ ∞) :
    IntegrableOn (fun x ↦ (μ.rnDeriv ν x).toReal) s ν 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {s : Set α} (hμs : μ s ≠ ⊤)
Goal: MeasureTheory.IntegrableOn (fun x => (μ.rnDeriv ν x).toReal) s ν
-- MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity'
lemma setIntegral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ]
    {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {s : Set α} (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = ((ν.withDensity (μ.rnDeriv ν)) s).toReal
-- MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity
lemma setIntegral_toReal_rnDeriv_eq_withDensity [SigmaFinite μ] [SFinite ν] (s : Set α) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν) s).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SFinite ν] (s : Set α)
Goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = ((ν.withDensity (μ.rnDeriv ν)) s).toReal
-- MeasureTheory.Measure.setIntegral_toReal_rnDeriv_le
lemma setIntegral_toReal_rnDeriv_le [SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ∞) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] {s : Set α} (hμs : μ s ≠ ⊤)
Goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal
-- MeasureTheory.Measure.setIntegral_toReal_rnDeriv'
lemma setIntegral_toReal_rnDeriv' [SigmaFinite μ] [HaveLebesgueDecomposition μ ν]
    (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) {s : Set α} (hs : MeasurableSet s)
Goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal
-- MeasureTheory.Measure.setIntegral_toReal_rnDeriv
lemma setIntegral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν) (s : Set α)
Goal: ∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν = (μ s).toReal
-- MeasureTheory.Measure.integral_toReal_rnDeriv
lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
    ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] (hμν : μ ≪ ν)
Goal: ∫ (x : α), (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal
-- MeasureTheory.Measure.rnDeriv_mul_rnDeriv
lemma rnDeriv_mul_rnDeriv {κ : Measure α} [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite κ]
    (hμν : μ ≪ ν) :
    μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν κ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite κ] (hμν : μ ≪ ν)
Goal: μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ
-- MeasureTheory.Measure.rnDeriv_le_one_of_le
lemma rnDeriv_le_one_of_le (hμν : μ ≤ ν) [SigmaFinite ν] : μ.rnDeriv ν ≤ᵐ[ν] 1 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (hμν : μ ≤ ν) [MeasureTheory.SigmaFinite ν]
Goal: μ.rnDeriv ν ≤ᵐ[ν] 1
-- MeasurableEmbedding.rnDeriv_map_aux
lemma _root_.MeasurableEmbedding.rnDeriv_map_aux (hf : MeasurableEmbedding f)
    (hμν : μ ≪ ν) [SigmaFinite μ] [SigmaFinite ν] :
    (fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (hμν : μ ≪ ν) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: (fun x => (MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν) (f x)) =ᵐ[ν] μ.rnDeriv ν
-- MeasurableEmbedding.rnDeriv_map
lemma _root_.MeasurableEmbedding.rnDeriv_map (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (fun x ↦ (μ.map f).rnDeriv (ν.map f) (f x)) =ᵐ[ν] μ.rnDeriv ν 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: (fun x => (MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν) (f x)) =ᵐ[ν] μ.rnDeriv ν
-- MeasurableEmbedding.map_withDensity_rnDeriv
lemma _root_.MeasurableEmbedding.map_withDensity_rnDeriv (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (ν.withDensity (μ.rnDeriv ν)).map f = (ν.map f).withDensity ((μ.map f).rnDeriv (ν.map f)) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: MeasureTheory.Measure.map f (ν.withDensity (μ.rnDeriv ν)) =
  (MeasureTheory.Measure.map f ν).withDensity ((MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν))
-- MeasurableEmbedding.singularPart_map
lemma _root_.MeasurableEmbedding.singularPart_map (hf : MeasurableEmbedding f)
    (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    (μ.map f).singularPart (ν.map f) = (μ.singularPart ν).map f 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β} (hf : MeasurableEmbedding f) (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: (MeasureTheory.Measure.map f μ).singularPart (MeasureTheory.Measure.map f ν) =
  MeasureTheory.Measure.map f (μ.singularPart ν)
-- MeasureTheory.SignedMeasure.withDensityᵥ_rnDeriv_eq
theorem withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α) [SigmaFinite μ]
    (h : s ≪ᵥ μ.toENNRealVectorMeasure) : μ.withDensityᵥ (s.rnDeriv μ) = s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (h : s ≪ᵥ μ.toENNRealVectorMeasure)
Goal: μ.withDensityᵥ (s.rnDeriv μ) = s
-- MeasureTheory.SignedMeasure.absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq
theorem absolutelyContinuous_iff_withDensityᵥ_rnDeriv_eq (s : SignedMeasure α) (μ : Measure α)
    [SigmaFinite μ] : s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s : MeasureTheory.SignedMeasure α) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s
-- MeasureTheory.lintegral_rnDeriv_mul
theorem lintegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) : ∫⁻ x, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞} (hf : AEMeasurable f ν)
Goal: ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLIntegral_rnDeriv_mul
lemma setLIntegral_rnDeriv_mul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f ν) {s : Set α} (hs : MeasurableSet s) :
    ∫⁻ x in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) {f : α → ℝ≥0∞} (hf : AEMeasurable f ν) {s : Set α} (hs : MeasurableSet s)
Goal: ∫⁻ (x : α) in s, μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.integrable_rnDeriv_smul_iff
theorem integrable_rnDeriv_smul_iff [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} :
    Integrable (fun x ↦ (μ.rnDeriv ν x).toReal • f x) ν ↔ Integrable f μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) [MeasureTheory.SigmaFinite μ] {f : α → E}
Goal: MeasureTheory.Integrable (fun x => (μ.rnDeriv ν x).toReal • f x) ν ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.withDensityᵥ_rnDeriv_smul
theorem withDensityᵥ_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} (hf : Integrable f μ) :
    ν.withDensityᵥ (fun x ↦ (rnDeriv μ ν x).toReal • f x) = μ.withDensityᵥ f 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) [MeasureTheory.SigmaFinite μ] {f : α → E} (hf : MeasureTheory.Integrable f μ)
Goal: (ν.withDensityᵥ fun x => (μ.rnDeriv ν x).toReal • f x) = μ.withDensityᵥ f
-- MeasureTheory.integral_rnDeriv_smul
theorem integral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
    [SigmaFinite μ] {f : α → E} :
    ∫ x, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x, f x ∂μ 
Arguments: {α : Type u_3} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [μ.HaveLebesgueDecomposition ν] (hμν : μ ≪ ν) [MeasureTheory.SigmaFinite μ] {f : α → E}
Goal: ∫ (x : α), (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ (x : α), f x ∂μ
-- hasFPowerSeriesWithinOnBall_univ
@[simp] lemma hasFPowerSeriesWithinOnBall_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {x : E} {r : ℝ≥0∞} :
    HasFPowerSeriesWithinOnBall f p univ x r ↔ HasFPowerSeriesOnBall f p x r 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞}
Goal: HasFPowerSeriesWithinOnBall f p Set.univ x r ↔ HasFPowerSeriesOnBall f p x r
-- hasFPowerSeriesWithinAt_univ
@[simp] lemma hasFPowerSeriesWithinAt_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} :
    HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E}
Goal: HasFPowerSeriesWithinAt f p Set.univ x ↔ HasFPowerSeriesAt f p x
-- analyticWithinAt_univ
@[simp] lemma analyticWithinAt_univ {f : E → F} {x : E} :
    AnalyticWithinAt 𝕜 f univ x ↔ AnalyticAt 𝕜 f x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E}
Goal: AnalyticWithinAt 𝕜 f Set.univ x ↔ AnalyticAt 𝕜 f x
-- analyticWithinOn_univ
lemma analyticWithinOn_univ {f : E → F} :
    AnalyticWithinOn 𝕜 f univ ↔ AnalyticOn 𝕜 f univ 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F}
Goal: AnalyticWithinOn 𝕜 f Set.univ ↔ AnalyticOn 𝕜 f Set.univ
-- HasFPowerSeriesWithinAt.continuousWithinAt
lemma HasFPowerSeriesWithinAt.continuousWithinAt {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
    {s : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p s x) : ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p s x)
Goal: ContinuousWithinAt f s x
-- AnalyticWithinAt.continuousWithinAt
lemma AnalyticWithinAt.continuousWithinAt {f : E → F} {s : Set E} {x : E}
    (h : AnalyticWithinAt 𝕜 f s x) : ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f s x)
Goal: ContinuousWithinAt f s x
-- analyticWithinAt_of_singleton_mem
lemma analyticWithinAt_of_singleton_mem {f : E → F} {s : Set E} {x : E} (h : {x} ∈ 𝓝[s] x) :
    AnalyticWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : {x} ∈ 𝓝[s] x)
Goal: AnalyticWithinAt 𝕜 f s x