ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.map_withDensity_abs_det_fderiv_eq_addHaar
theorem map_withDensity_abs_det_fderiv_eq_addHaar (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (h'f : Measurable f) :
    Measure.map f ((μ.restrict s).withDensity fun x => ENNReal.ofReal |(f' x).det|) =
      μ.restrict (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (h'f : Measurable f)
Goal: MeasureTheory.Measure.map f ((μ.restrict s).withDensity fun x => ENNReal.ofReal |(f' x).det|) = μ.restrict (f '' s)
-- MeasureTheory.restrict_map_withDensity_abs_det_fderiv_eq_addHaar
theorem restrict_map_withDensity_abs_det_fderiv_eq_addHaar (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
    Measure.map (s.restrict f) (comap (↑) (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
      μ.restrict (f '' s) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s)
Goal: MeasureTheory.Measure.map (s.restrict f)
    (MeasureTheory.Measure.comap Subtype.val (μ.withDensity fun x => ENNReal.ofReal |(f' x).det|)) =
  μ.restrict (f '' s)
-- MeasureTheory.lintegral_image_eq_lintegral_abs_det_fderiv_mul
theorem lintegral_image_eq_lintegral_abs_det_fderiv_mul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → ℝ≥0∞) :
    ∫⁻ x in f '' s, g x ∂μ = ∫⁻ x in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (g : E → ℝ≥0∞)
Goal: ∫⁻ (x : E) in f '' s, g x ∂μ = ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| * g (f x) ∂μ
-- MeasureTheory.integrableOn_image_iff_integrableOn_abs_det_fderiv_smul
theorem integrableOn_image_iff_integrableOn_abs_det_fderiv_smul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) :
    IntegrableOn g (f '' s) μ ↔ IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (g : E → F)
Goal: MeasureTheory.IntegrableOn g (f '' s) μ ↔ MeasureTheory.IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ
-- MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul
theorem integral_image_eq_integral_abs_det_fderiv_smul (hs : MeasurableSet s)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) :
    ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (g : E → F)
Goal: ∫ (x : E) in f '' s, g x ∂μ = ∫ (x : E) in s, |(f' x).det| • g (f x) ∂μ
-- MeasureTheory.det_one_smulRight
theorem det_one_smulRight {𝕜 : Type*} [NormedField 𝕜] (v : 𝕜) :
    ((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v 
Arguments: {𝕜 : Type u_3} [NormedField 𝕜] (v : 𝕜)
Goal: (ContinuousLinearMap.smulRight 1 v).det = v
-- MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul
theorem integrableOn_image_iff_integrableOn_abs_deriv_smul {s : Set ℝ} {f : ℝ → ℝ} {f' : ℝ → ℝ}
    (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (hf : InjOn f s)
    (g : ℝ → F) : IntegrableOn g (f '' s) ↔ IntegrableOn (fun x => |f' x| • g (f x)) s 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set ℝ} {f f' : ℝ → ℝ} (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (g : ℝ → F)
Goal: MeasureTheory.IntegrableOn g (f '' s) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn (fun x => |f' x| • g (f x)) s MeasureTheory.volume
-- MeasureTheory.integral_image_eq_integral_abs_deriv_smul
theorem integral_image_eq_integral_abs_deriv_smul {s : Set ℝ} {f : ℝ → ℝ} {f' : ℝ → ℝ}
    (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
    (hf : InjOn f s) (g : ℝ → F) : ∫ x in f '' s, g x = ∫ x in s, |f' x| • g (f x) 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set ℝ} {f f' : ℝ → ℝ} (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (hf : Set.InjOn f s) (g : ℝ → F)
Goal: ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)
-- MeasureTheory.integral_target_eq_integral_abs_det_fderiv_smul
theorem integral_target_eq_integral_abs_det_fderiv_smul {f : PartialHomeomorph E E}
    (hf' : ∀ x ∈ f.source, HasFDerivAt f (f' x) x) (g : E → F) :
    ∫ x in f.target, g x ∂μ = ∫ x in f.source, |(f' x).det| • g (f x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {f : PartialHomeomorph E E} (hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x) (g : E → F)
Goal: ∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ
-- MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul
lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul
    (hs : MeasurableSet s) (hf : MeasurableEmbedding f)
    {g : E → ℝ} (hg : ∀ᵐ x ∂μ, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s) μ)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    (μ.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (hf : MeasurableEmbedding f) {g : E → ℝ} (hg : ∀ᵐ (x : E) ∂μ, x ∈ f '' s → 0 ≤ g x) (hg_int : MeasureTheory.IntegrableOn g (f '' s) μ) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
Goal: (MeasureTheory.Measure.comap f (μ.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)
-- MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul
lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_det_fderiv_mul
    (hs : MeasurableSet s) (f : E ≃ᵐ E)
    {g : E → ℝ} (hg : ∀ᵐ x ∂μ, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s) μ)
    (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
    (μ.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (hs : MeasurableSet s) (f : E ≃ᵐ E) {g : E → ℝ} (hg : ∀ᵐ (x : E) ∂μ, x ∈ ⇑f '' s → 0 ≤ g x) (hg_int : MeasureTheory.IntegrableOn g (⇑f '' s) μ) (hf' : ∀ x ∈ s, HasFDerivWithinAt (⇑f) (f' x) s x)
Goal: (MeasureTheory.Measure.map (⇑f.symm) (μ.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)
-- MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul
lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul
    {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s)
    {g : ℝ → ℝ} (hg : ∀ᵐ x, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s))
    {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
Arguments: {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s) {g : ℝ → ℝ} (hg : ∀ᵐ (x : ℝ), x ∈ f '' s → 0 ≤ g x) (hg_int : MeasureTheory.IntegrableOn g (f '' s) MeasureTheory.volume) {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x)
Goal: (MeasureTheory.Measure.comap f (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))
-- MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul
lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul
    (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s)
    {g : ℝ → ℝ} (hg : ∀ᵐ x, x ∈ f '' s → 0 ≤ g x) (hg_int : IntegrableOn g (f '' s))
    {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
Arguments: (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s) {g : ℝ → ℝ} (hg : ∀ᵐ (x : ℝ), x ∈ ⇑f '' s → 0 ≤ g x) (hg_int : MeasureTheory.IntegrableOn g (⇑f '' s) MeasureTheory.volume) {f' : ℝ → ℝ} (hf' : ∀ x ∈ s, HasDerivWithinAt (⇑f) (f' x) s x)
Goal: (MeasureTheory.Measure.map (⇑f.symm) (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))
-- MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'
lemma _root_.MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'
    {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s)
    {f' : ℝ → ℝ} (hf' : ∀ x, HasDerivAt f (f' x) x)
    {g : ℝ → ℝ} (hg : 0 ≤ᵐ[volume] g) (hg_int : Integrable g) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).comap f s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
Arguments: {f : ℝ → ℝ} (hf : MeasurableEmbedding f) {s : Set ℝ} (hs : MeasurableSet s) {f' : ℝ → ℝ} (hf' : ∀ (x : ℝ), HasDerivAt f (f' x) x) {g : ℝ → ℝ} (hg : 0 ≤ᵐ[MeasureTheory.volume] g) (hg_int : MeasureTheory.Integrable g MeasureTheory.volume)
Goal: (MeasureTheory.Measure.comap f (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))
-- MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'
lemma _root_.MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'
    (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s)
    {f' : ℝ → ℝ} (hf' : ∀ x, HasDerivAt f (f' x) x)
    {g : ℝ → ℝ} (hg : 0 ≤ᵐ[volume] g) (hg_int : Integrable g) :
    (volume.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
      = ENNReal.ofReal (∫ x in s, |f' x| * g (f x)) 
Arguments: (f : ℝ ≃ᵐ ℝ) {s : Set ℝ} (hs : MeasurableSet s) {f' : ℝ → ℝ} (hf' : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x) {g : ℝ → ℝ} (hg : 0 ≤ᵐ[MeasureTheory.volume] g) (hg_int : MeasureTheory.Integrable g MeasureTheory.volume)
Goal: (MeasureTheory.Measure.map (⇑f.symm) (MeasureTheory.volume.withDensity fun x => ENNReal.ofReal (g x))) s =
  ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * g (f x))
-- MeasureTheory.eLpNorm_aeeqFun
@[simp]
theorem eLpNorm_aeeqFun {α E : Type*} [MeasurableSpace α] {μ : Measure α} [NormedAddCommGroup E]
    {p : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ) :
    eLpNorm (AEEqFun.mk f hf) p μ = eLpNorm f p μ 
Arguments: {α : Type u_5} {E : Type u_6} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.eLpNorm (↑(MeasureTheory.AEEqFun.mk f hf)) p μ = MeasureTheory.eLpNorm f p μ
-- MeasureTheory.Memℒp.eLpNorm_mk_lt_top
theorem Memℒp.eLpNorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
    [NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
    eLpNorm (AEEqFun.mk f hfp.1) p μ < ∞ 
Arguments: {α : Type u_5} {E : Type u_6} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.eLpNorm (↑(MeasureTheory.AEEqFun.mk f ⋯)) p μ < ⊤
-- MeasureTheory.Memℒp.toLp_val
theorem toLp_val {f : α → E} (h : Memℒp f p μ) : (toLp f h).1 = AEEqFun.mk f h.1 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (h : MeasureTheory.Memℒp f p μ)
Goal: ↑(MeasureTheory.Memℒp.toLp f h) = MeasureTheory.AEEqFun.mk f ⋯
-- MeasureTheory.Memℒp.coeFn_toLp
theorem coeFn_toLp {f : α → E} (hf : Memℒp f p μ) : hf.toLp f =ᵐ[μ] f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: ↑↑(MeasureTheory.Memℒp.toLp f hf) =ᵐ[μ] f
-- MeasureTheory.Memℒp.toLp_congr
theorem toLp_congr {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) (hfg : f =ᵐ[μ] g) :
    hf.toLp f = hg.toLp g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ) (hfg : f =ᵐ[μ] g)
Goal: MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp g hg
-- MeasureTheory.Memℒp.toLp_eq_toLp_iff
@[simp]
theorem toLp_eq_toLp_iff {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    hf.toLp f = hg.toLp g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp g hg ↔ f =ᵐ[μ] g
-- MeasureTheory.Memℒp.toLp_zero
@[simp]
theorem toLp_zero (h : Memℒp (0 : α → E) p μ) : h.toLp 0 = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (h : MeasureTheory.Memℒp 0 p μ)
Goal: MeasureTheory.Memℒp.toLp 0 h = 0
-- MeasureTheory.Memℒp.toLp_add
theorem toLp_add {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    (hf.add hg).toLp (f + g) = hf.toLp f + hg.toLp g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp.toLp (f + g) ⋯ = MeasureTheory.Memℒp.toLp f hf + MeasureTheory.Memℒp.toLp g hg
-- MeasureTheory.Memℒp.toLp_neg
theorem toLp_neg {f : α → E} (hf : Memℒp f p μ) : hf.neg.toLp (-f) = -hf.toLp f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: MeasureTheory.Memℒp.toLp (-f) ⋯ = -MeasureTheory.Memℒp.toLp f hf
-- MeasureTheory.Memℒp.toLp_sub
theorem toLp_sub {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    (hf.sub hg).toLp (f - g) = hf.toLp f - hg.toLp g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp.toLp (f - g) ⋯ = MeasureTheory.Memℒp.toLp f hf - MeasureTheory.Memℒp.toLp g hg
-- MeasureTheory.Lp.ext
@[ext high]
theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : ↥(MeasureTheory.Lp E p μ)} (h : ↑↑f =ᵐ[μ] ↑↑g)
Goal: f = g
-- MeasureTheory.Lp.ext_iff
theorem ext_iff {f g : Lp E p μ} : f = g ↔ f =ᵐ[μ] g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f g : ↥(MeasureTheory.Lp E p μ)}
Goal: f = g ↔ ↑↑f =ᵐ[μ] ↑↑g
-- MeasureTheory.Lp.mem_Lp_iff_eLpNorm_lt_top
theorem mem_Lp_iff_eLpNorm_lt_top {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ eLpNorm f p μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E}
Goal: f ∈ MeasureTheory.Lp E p μ ↔ MeasureTheory.eLpNorm (↑f) p μ < ⊤
-- MeasureTheory.Lp.mem_Lp_iff_memℒp
theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E}
Goal: f ∈ MeasureTheory.Lp E p μ ↔ MeasureTheory.Memℒp (↑f) p μ
-- MeasureTheory.Lp.antitone
protected theorem antitone [IsFiniteMeasure μ] {p q : ℝ≥0∞} (hpq : p ≤ q) : Lp E q μ ≤ Lp E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [MeasureTheory.IsFiniteMeasure μ] {p q : ℝ≥0∞} (hpq : p ≤ q)
Goal: MeasureTheory.Lp E q μ ≤ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.coeFn_mk
@[simp]
theorem coeFn_mk {f : α →ₘ[μ] E} (hf : eLpNorm f p μ < ∞) : ((⟨f, hf⟩ : Lp E p μ) : α → E) = f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} (hf : MeasureTheory.eLpNorm (↑f) p μ < ⊤)
Goal: ↑↑⟨f, hf⟩ = ↑f
-- MeasureTheory.Lp.coe_mk
theorem coe_mk {f : α →ₘ[μ] E} (hf : eLpNorm f p μ < ∞) : ((⟨f, hf⟩ : Lp E p μ) : α →ₘ[μ] E) = f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α →ₘ[μ] E} (hf : MeasureTheory.eLpNorm (↑f) p μ < ⊤)
Goal: ↑⟨f, hf⟩ = f
-- MeasureTheory.Lp.toLp_coeFn
@[simp]
theorem toLp_coeFn (f : Lp E p μ) (hf : Memℒp f p μ) : hf.toLp f = f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ)) (hf : MeasureTheory.Memℒp (↑↑f) p μ)
Goal: MeasureTheory.Memℒp.toLp (↑↑f) hf = f
-- MeasureTheory.Lp.eLpNorm_lt_top
theorem eLpNorm_lt_top (f : Lp E p μ) : eLpNorm f p μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.eLpNorm (↑↑f) p μ < ⊤
-- MeasureTheory.Lp.eLpNorm_ne_top
theorem eLpNorm_ne_top (f : Lp E p μ) : eLpNorm f p μ ≠ ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.eLpNorm (↑↑f) p μ ≠ ⊤
-- MeasureTheory.Lp.stronglyMeasurable
@[measurability]
protected theorem stronglyMeasurable (f : Lp E p μ) : StronglyMeasurable f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.StronglyMeasurable ↑↑f
-- MeasureTheory.Lp.aestronglyMeasurable
@[measurability]
protected theorem aestronglyMeasurable (f : Lp E p μ) : AEStronglyMeasurable f μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.AEStronglyMeasurable (↑↑f) μ
-- MeasureTheory.Lp.memℒp
protected theorem memℒp (f : Lp E p μ) : Memℒp f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: MeasureTheory.Memℒp (↑↑f) p μ
-- MeasureTheory.Lp.coeFn_zero
theorem coeFn_zero : ⇑(0 : Lp E p μ) =ᵐ[μ] 0 
Arguments: {α : Type u_1} (E : Type u_2) {m0 : MeasurableSpace α} (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E]
Goal: ↑↑0 =ᵐ[μ] 0
-- MeasureTheory.Lp.coeFn_neg
theorem coeFn_neg (f : Lp E p μ) : ⇑(-f) =ᵐ[μ] -f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(-f) =ᵐ[μ] -↑↑f
-- MeasureTheory.Lp.coeFn_add
theorem coeFn_add (f g : Lp E p μ) : ⇑(f + g) =ᵐ[μ] f + g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(f + g) =ᵐ[μ] ↑↑f + ↑↑g
-- MeasureTheory.Lp.coeFn_sub
theorem coeFn_sub (f g : Lp E p μ) : ⇑(f - g) =ᵐ[μ] f - g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f g : ↥(MeasureTheory.Lp E p μ))
Goal: ↑↑(f - g) =ᵐ[μ] ↑↑f - ↑↑g
-- MeasureTheory.Lp.const_mem_Lp
theorem const_mem_Lp (α) {_ : MeasurableSpace α} (μ : Measure α) (c : E) [IsFiniteMeasure μ] :
    @AEEqFun.const α _ _ μ _ c ∈ Lp E p μ 
Arguments: {E : Type u_2} {p : ℝ≥0∞} [NormedAddCommGroup E] (α : Type u_5)
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (c : E) [inst : MeasureTheory.IsFiniteMeasure μ],
  MeasureTheory.AEEqFun.const α c ∈ MeasureTheory.Lp E p μ
-- MeasureTheory.Lp.norm_def
theorem norm_def (f : Lp E p μ) : ‖f‖ = ENNReal.toReal (eLpNorm f p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖f‖ = (MeasureTheory.eLpNorm (↑↑f) p μ).toReal
-- MeasureTheory.Lp.nnnorm_def
theorem nnnorm_def (f : Lp E p μ) : ‖f‖₊ = ENNReal.toNNReal (eLpNorm f p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ‖f‖₊ = (MeasureTheory.eLpNorm (↑↑f) p μ).toNNReal
-- MeasureTheory.Lp.coe_nnnorm
@[simp, norm_cast]
protected theorem coe_nnnorm (f : Lp E p μ) : (‖f‖₊ : ℝ) = ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑‖f‖₊ = ‖f‖
-- MeasureTheory.Lp.nnnorm_coe_ennreal
@[simp, norm_cast]
theorem nnnorm_coe_ennreal (f : Lp E p μ) : (‖f‖₊ : ℝ≥0∞) = eLpNorm f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑‖f‖₊ = MeasureTheory.eLpNorm (↑↑f) p μ
-- MeasureTheory.Lp.norm_toLp
@[simp]
lemma norm_toLp (f : α → E) (hf : Memℒp f p μ) : ‖hf.toLp f‖ = ENNReal.toReal (eLpNorm f p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : α → E) (hf : MeasureTheory.Memℒp f p μ)
Goal: ‖MeasureTheory.Memℒp.toLp f hf‖ = (MeasureTheory.eLpNorm f p μ).toReal
-- MeasureTheory.Lp.nnnorm_toLp
@[simp]
theorem nnnorm_toLp (f : α → E) (hf : Memℒp f p μ) :
    ‖hf.toLp f‖₊ = ENNReal.toNNReal (eLpNorm f p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] (f : α → E) (hf : MeasureTheory.Memℒp f p μ)
Goal: ‖MeasureTheory.Memℒp.toLp f hf‖₊ = (MeasureTheory.eLpNorm f p μ).toNNReal
-- MeasureTheory.Lp.coe_nnnorm_toLp
theorem coe_nnnorm_toLp {f : α → E} (hf : Memℒp f p μ) : (‖hf.toLp f‖₊ : ℝ≥0∞) = eLpNorm f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] {f : α → E} (hf : MeasureTheory.Memℒp f p μ)
Goal: ↑‖MeasureTheory.Memℒp.toLp f hf‖₊ = MeasureTheory.eLpNorm f p μ