ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearPMap.adjoint_apply_eq
theorem adjoint_apply_eq (y : T†.domain) {x₀ : E} (hx₀ : ∀ x : T.domain, ⟪x₀, x⟫ = ⟪(y : F), T x⟫) :
    T† y = x₀ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] {T : E →ₗ.[𝕜] F} (hT : Dense ↑T.domain) [CompleteSpace E] (y : ↥T.adjoint.domain) {x₀ : E} (hx₀ : ∀ (x : ↥T.domain), ⟪x₀, ↑x⟫_𝕜 = ⟪↑y, ↑T x⟫_𝕜)
Goal: ↑T.adjoint y = x₀
-- LinearPMap.adjoint_isFormalAdjoint
theorem adjoint_isFormalAdjoint : T†.IsFormalAdjoint T 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] {T : E →ₗ.[𝕜] F} (hT : Dense ↑T.domain) [CompleteSpace E]
Goal: T.adjoint.IsFormalAdjoint T
-- LinearPMap.IsFormalAdjoint.le_adjoint
theorem IsFormalAdjoint.le_adjoint (h : T.IsFormalAdjoint S) : S ≤ T† 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] {T : E →ₗ.[𝕜] F} {S : F →ₗ.[𝕜] E} (hT : Dense ↑T.domain) [CompleteSpace E] (h : T.IsFormalAdjoint S)
Goal: S ≤ T.adjoint
-- ContinuousLinearMap.toPMap_adjoint_eq_adjoint_toPMap_of_dense
theorem toPMap_adjoint_eq_adjoint_toPMap_of_dense (hp : Dense (p : Set E)) :
    (A.toPMap p).adjoint = A.adjoint.toPMap ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) {p : Submodule 𝕜 E} (hp : Dense ↑p)
Goal: ((↑A).toPMap p).adjoint = (↑(ContinuousLinearMap.adjoint A)).toPMap ⊤
-- LinearPMap.isSelfAdjoint_def
theorem isSelfAdjoint_def : IsSelfAdjoint A ↔ A† = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →ₗ.[𝕜] E}
Goal: IsSelfAdjoint A ↔ A.adjoint = A
-- IsSelfAdjoint.dense_domain
theorem _root_.IsSelfAdjoint.dense_domain (hA : IsSelfAdjoint A) : Dense (A.domain : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A : E →ₗ.[𝕜] E} (hA : IsSelfAdjoint A)
Goal: Dense ↑A.domain
-- MeasureTheory.weightedSMul_apply
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
    weightedSMul μ s x = (μ s).toReal • x 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α) (x : F)
Goal: (MeasureTheory.weightedSMul μ s) x = (μ s).toReal • x
-- MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
    weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α}
Goal: MeasureTheory.weightedSMul 0 = 0
-- MeasureTheory.weightedSMul_empty
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
    weightedSMul μ ∅ = (0 : F →L[ℝ] F) 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.weightedSMul μ ∅ = 0
-- MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
    (hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
    (weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) {s : Set α} (hμs : μ s ≠ ⊤) (hνs : ν s ≠ ⊤)
Goal: MeasureTheory.weightedSMul (μ + ν) s = MeasureTheory.weightedSMul μ s + MeasureTheory.weightedSMul ν s
-- MeasureTheory.weightedSMul_smul_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
    (weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (c : ℝ≥0∞) {s : Set α}
Goal: MeasureTheory.weightedSMul (c • μ) s = c.toReal • MeasureTheory.weightedSMul μ s
-- MeasureTheory.weightedSMul_congr
theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
    (weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α) (hst : μ s = μ t)
Goal: MeasureTheory.weightedSMul μ s = MeasureTheory.weightedSMul μ t
-- MeasureTheory.weightedSMul_null
theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (h_zero : μ s = 0)
Goal: MeasureTheory.weightedSMul μ s = 0
-- MeasureTheory.weightedSMul_union'
theorem weightedSMul_union' (s t : Set α) (ht : MeasurableSet t) (hs_finite : μ s ≠ ∞)
    (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
    (weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α) (ht : MeasurableSet t) (hs_finite : μ s ≠ ⊤) (ht_finite : μ t ≠ ⊤) (h_inter : s ∩ t = ∅)
Goal: MeasureTheory.weightedSMul μ (s ∪ t) = MeasureTheory.weightedSMul μ s + MeasureTheory.weightedSMul μ t
-- MeasureTheory.weightedSMul_union
@[nolint unusedArguments]
theorem weightedSMul_union (s t : Set α) (_hs : MeasurableSet s) (ht : MeasurableSet t)
    (hs_finite : μ s ≠ ∞) (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
    (weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t : Set α) (_hs : MeasurableSet s) (ht : MeasurableSet t) (hs_finite : μ s ≠ ⊤) (ht_finite : μ t ≠ ⊤) (h_inter : s ∩ t = ∅)
Goal: MeasureTheory.weightedSMul μ (s ∪ t) = MeasureTheory.weightedSMul μ s + MeasureTheory.weightedSMul μ t
-- MeasureTheory.weightedSMul_smul
theorem weightedSMul_smul [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜)
    (s : Set α) (x : F) : weightedSMul μ s (c • x) = c • weightedSMul μ s x 
Arguments: {α : Type u_1} {F : Type u_3} {𝕜 : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜) (s : Set α) (x : F)
Goal: (MeasureTheory.weightedSMul μ s) (c • x) = c • (MeasureTheory.weightedSMul μ s) x
-- MeasureTheory.norm_weightedSMul_le
theorem norm_weightedSMul_le (s : Set α) : ‖(weightedSMul μ s : F →L[ℝ] F)‖ ≤ (μ s).toReal 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α)
Goal: ‖MeasureTheory.weightedSMul μ s‖ ≤ (μ s).toReal
-- MeasureTheory.dominatedFinMeasAdditive_weightedSMul
theorem dominatedFinMeasAdditive_weightedSMul {_ : MeasurableSpace α} (μ : Measure α) :
    DominatedFinMeasAdditive μ (weightedSMul μ : Set α → F →L[ℝ] F) 1 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α),
  MeasureTheory.DominatedFinMeasAdditive μ (MeasureTheory.weightedSMul μ) 1
-- MeasureTheory.weightedSMul_nonneg
theorem weightedSMul_nonneg (s : Set α) (x : ℝ) (hx : 0 ≤ x) : 0 ≤ weightedSMul μ s x 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Set α) (x : ℝ) (hx : 0 ≤ x)
Goal: 0 ≤ (MeasureTheory.weightedSMul μ s) x
-- MeasureTheory.SimpleFunc.posPart_map_norm
theorem posPart_map_norm (f : α →ₛ ℝ) : (posPart f).map norm = posPart f 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ)
Goal: MeasureTheory.SimpleFunc.map norm f.posPart = f.posPart
-- MeasureTheory.SimpleFunc.negPart_map_norm
theorem negPart_map_norm (f : α →ₛ ℝ) : (negPart f).map norm = negPart f 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ)
Goal: MeasureTheory.SimpleFunc.map norm f.negPart = f.negPart
-- MeasureTheory.SimpleFunc.posPart_sub_negPart
theorem posPart_sub_negPart (f : α →ₛ ℝ) : f.posPart - f.negPart = f 
Arguments: {α : Type u_1} [MeasurableSpace α] (f : MeasureTheory.SimpleFunc α ℝ)
Goal: f.posPart - f.negPart = f
-- MeasureTheory.SimpleFunc.integral_def
theorem integral_def {_ : MeasurableSpace α} (μ : Measure α) (f : α →ₛ F) :
    f.integral μ = f.setToSimpleFunc (weightedSMul μ) 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : MeasureTheory.SimpleFunc α F),
  MeasureTheory.SimpleFunc.integral μ f = MeasureTheory.SimpleFunc.setToSimpleFunc (MeasureTheory.weightedSMul μ) f
-- MeasureTheory.SimpleFunc.integral_eq
theorem integral_eq {m : MeasurableSpace α} (μ : Measure α) (f : α →ₛ F) :
    f.integral μ = ∑ x ∈ f.range, (μ (f ⁻¹' {x})).toReal • x 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : MeasureTheory.SimpleFunc α F)
Goal: MeasureTheory.SimpleFunc.integral μ f = ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • x
-- MeasureTheory.SimpleFunc.integral_eq_sum_filter
theorem integral_eq_sum_filter [DecidablePred fun x : F => x ≠ 0] {m : MeasurableSpace α}
    (f : α →ₛ F) (μ : Measure α) :
    f.integral μ = ∑ x ∈ f.range.filter fun x => x ≠ 0, (μ (f ⁻¹' {x})).toReal • x 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] [DecidablePred fun x => x ≠ 0] {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α F) (μ : MeasureTheory.Measure α)
Goal: MeasureTheory.SimpleFunc.integral μ f = ∑ x ∈ Finset.filter (fun x => x ≠ 0) f.range, (μ (↑f ⁻¹' {x})).toReal • x
-- MeasureTheory.SimpleFunc.integral_eq_sum_of_subset
theorem integral_eq_sum_of_subset [DecidablePred fun x : F => x ≠ 0] {f : α →ₛ F} {s : Finset F}
    (hs : (f.range.filter fun x => x ≠ 0) ⊆ s) :
    f.integral μ = ∑ x ∈ s, (μ (f ⁻¹' {x})).toReal • x 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [DecidablePred fun x => x ≠ 0] {f : MeasureTheory.SimpleFunc α F} {s : Finset F} (hs : Finset.filter (fun x => x ≠ 0) f.range ⊆ s)
Goal: MeasureTheory.SimpleFunc.integral μ f = ∑ x ∈ s, (μ (↑f ⁻¹' {x})).toReal • x
-- MeasureTheory.SimpleFunc.integral_const
@[simp]
theorem integral_const {m : MeasurableSpace α} (μ : Measure α) (y : F) :
    (const α y).integral μ = (μ univ).toReal • y 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) (y : F)
Goal: MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.const α y) = (μ Set.univ).toReal • y
-- MeasureTheory.SimpleFunc.integral_piecewise_zero
@[simp]
theorem integral_piecewise_zero {m : MeasurableSpace α} (f : α →ₛ F) (μ : Measure α) {s : Set α}
    (hs : MeasurableSet s) : (piecewise s hs f 0).integral μ = f.integral (μ.restrict s) 
Arguments: {α : Type u_1} {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} (f : MeasureTheory.SimpleFunc α F) (μ : MeasureTheory.Measure α) {s : Set α} (hs : MeasurableSet s)
Goal: MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.piecewise s hs f 0) =
  MeasureTheory.SimpleFunc.integral (μ.restrict s) f
-- MeasureTheory.SimpleFunc.map_integral
theorem map_integral (f : α →ₛ E) (g : E → F) (hf : Integrable f μ) (hg : g 0 = 0) :
    (f.map g).integral μ = ∑ x ∈ f.range, ENNReal.toReal (μ (f ⁻¹' {x})) • g x 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (g : E → F) (hf : MeasureTheory.Integrable (↑f) μ) (hg : g 0 = 0)
Goal: MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map g f) = ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • g x
-- MeasureTheory.SimpleFunc.integral_eq_lintegral'
theorem integral_eq_lintegral' {f : α →ₛ E} {g : E → ℝ≥0∞} (hf : Integrable f μ) (hg0 : g 0 = 0)
    (ht : ∀ b, g b ≠ ∞) :
    (f.map (ENNReal.toReal ∘ g)).integral μ = ENNReal.toReal (∫⁻ a, g (f a) ∂μ) 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α E} {g : E → ℝ≥0∞} (hf : MeasureTheory.Integrable (↑f) μ) (hg0 : g 0 = 0) (ht : ∀ (b : E), g b ≠ ⊤)
Goal: MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map (ENNReal.toReal ∘ g) f) =
  (∫⁻ (a : α), g (↑f a) ∂μ).toReal
-- MeasureTheory.SimpleFunc.integral_congr
theorem integral_congr {f g : α →ₛ E} (hf : Integrable f μ) (h : f =ᵐ[μ] g) :
    f.integral μ = g.integral μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (h : ↑f =ᵐ[μ] ↑g)
Goal: MeasureTheory.SimpleFunc.integral μ f = MeasureTheory.SimpleFunc.integral μ g
-- MeasureTheory.SimpleFunc.integral_eq_lintegral
theorem integral_eq_lintegral {f : α →ₛ ℝ} (hf : Integrable f μ) (h_pos : 0 ≤ᵐ[μ] f) :
    f.integral μ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal (f a) ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : MeasureTheory.SimpleFunc α ℝ} (hf : MeasureTheory.Integrable (↑f) μ) (h_pos : 0 ≤ᵐ[μ] ↑f)
Goal: MeasureTheory.SimpleFunc.integral μ f = (∫⁻ (a : α), ENNReal.ofReal (↑f a) ∂μ).toReal
-- MeasureTheory.SimpleFunc.integral_add
theorem integral_add {f g : α →ₛ E} (hf : Integrable f μ) (hg : Integrable g μ) :
    integral μ (f + g) = integral μ f + integral μ g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (hg : MeasureTheory.Integrable (↑g) μ)
Goal: MeasureTheory.SimpleFunc.integral μ (f + g) =
  MeasureTheory.SimpleFunc.integral μ f + MeasureTheory.SimpleFunc.integral μ g
-- MeasureTheory.SimpleFunc.integral_neg
theorem integral_neg {f : α →ₛ E} (hf : Integrable f μ) : integral μ (-f) = -integral μ f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.integral μ (-f) = -MeasureTheory.SimpleFunc.integral μ f
-- MeasureTheory.SimpleFunc.integral_sub
theorem integral_sub {f g : α →ₛ E} (hf : Integrable f μ) (hg : Integrable g μ) :
    integral μ (f - g) = integral μ f - integral μ g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {f g : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ) (hg : MeasureTheory.Integrable (↑g) μ)
Goal: MeasureTheory.SimpleFunc.integral μ (f - g) =
  MeasureTheory.SimpleFunc.integral μ f - MeasureTheory.SimpleFunc.integral μ g
-- MeasureTheory.SimpleFunc.integral_smul
theorem integral_smul (c : 𝕜) {f : α →ₛ E} (hf : Integrable f μ) :
    integral μ (c • f) = c • integral μ f 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [SMulCommClass ℝ 𝕜 E] (c : 𝕜) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: MeasureTheory.SimpleFunc.integral μ (c • f) = c • MeasureTheory.SimpleFunc.integral μ f
-- MeasureTheory.SimpleFunc.norm_setToSimpleFunc_le_integral_norm
theorem norm_setToSimpleFunc_le_integral_norm (T : Set α → E →L[ℝ] F) {C : ℝ}
    (hT_norm : ∀ s, MeasurableSet s → μ s < ∞ → ‖T s‖ ≤ C * (μ s).toReal) {f : α →ₛ E}
    (hf : Integrable f μ) : ‖f.setToSimpleFunc T‖ ≤ C * (f.map norm).integral μ 
Arguments: {α : Type u_1} {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (T : Set α → E →L[ℝ] F) {C : ℝ} (hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal) {f : MeasureTheory.SimpleFunc α E} (hf : MeasureTheory.Integrable (↑f) μ)
Goal: ‖MeasureTheory.SimpleFunc.setToSimpleFunc T f‖ ≤
  C * MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map norm f)
-- MeasureTheory.SimpleFunc.norm_integral_le_integral_norm
theorem norm_integral_le_integral_norm (f : α →ₛ E) (hf : Integrable f μ) :
    ‖f.integral μ‖ ≤ (f.map norm).integral μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) μ)
Goal: ‖MeasureTheory.SimpleFunc.integral μ f‖ ≤ MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map norm f)
-- MeasureTheory.SimpleFunc.integral_add_measure
theorem integral_add_measure {ν} (f : α →ₛ E) (hf : Integrable f (μ + ν)) :
    f.integral (μ + ν) = f.integral μ + f.integral ν 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {ν : MeasureTheory.Measure α} (f : MeasureTheory.SimpleFunc α E) (hf : MeasureTheory.Integrable (↑f) (μ + ν))
Goal: MeasureTheory.SimpleFunc.integral (μ + ν) f =
  MeasureTheory.SimpleFunc.integral μ f + MeasureTheory.SimpleFunc.integral ν f
-- MeasureTheory.L1.SimpleFunc.norm_eq_integral
theorem norm_eq_integral (f : α →₁ₛ[μ] E) : ‖f‖ = ((toSimpleFunc f).map norm).integral μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: ‖f‖ =
  MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map norm (MeasureTheory.Lp.simpleFunc.toSimpleFunc f))
-- MeasureTheory.L1.SimpleFunc.coe_posPart
@[norm_cast]
theorem coe_posPart (f : α →₁ₛ[μ] ℝ) : (posPart f : α →₁[μ] ℝ) = Lp.posPart (f : α →₁[μ] ℝ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ))
Goal: ↑(MeasureTheory.L1.SimpleFunc.posPart f) = MeasureTheory.Lp.posPart ↑f
-- MeasureTheory.L1.SimpleFunc.coe_negPart
@[norm_cast]
theorem coe_negPart (f : α →₁ₛ[μ] ℝ) : (negPart f : α →₁[μ] ℝ) = Lp.negPart (f : α →₁[μ] ℝ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ))
Goal: ↑(MeasureTheory.L1.SimpleFunc.negPart f) = MeasureTheory.Lp.negPart ↑f
-- MeasureTheory.L1.SimpleFunc.integral_eq_integral
theorem integral_eq_integral (f : α →₁ₛ[μ] E) : integral f = (toSimpleFunc f).integral μ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.integral f =
  MeasureTheory.SimpleFunc.integral μ (MeasureTheory.Lp.simpleFunc.toSimpleFunc f)
-- MeasureTheory.L1.SimpleFunc.integral_eq_lintegral
nonrec theorem integral_eq_lintegral {f : α →₁ₛ[μ] ℝ} (h_pos : 0 ≤ᵐ[μ] toSimpleFunc f) :
    integral f = ENNReal.toReal (∫⁻ a, ENNReal.ofReal ((toSimpleFunc f) a) ∂μ) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ)} (h_pos : 0 ≤ᵐ[μ] ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f))
Goal: MeasureTheory.L1.SimpleFunc.integral f =
  (∫⁻ (a : α), ENNReal.ofReal (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) a) ∂μ).toReal
-- MeasureTheory.L1.SimpleFunc.integral_eq_setToL1S
theorem integral_eq_setToL1S (f : α →₁ₛ[μ] E) : integral f = setToL1S (weightedSMul μ) f 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.integral f = MeasureTheory.L1.SimpleFunc.setToL1S (MeasureTheory.weightedSMul μ) f
-- MeasureTheory.L1.SimpleFunc.integral_congr
nonrec theorem integral_congr {f g : α →₁ₛ[μ] E} (h : toSimpleFunc f =ᵐ[μ] toSimpleFunc g) :
    integral f = integral g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] {f g : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)} (h : ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) =ᵐ[μ] ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc g))
Goal: MeasureTheory.L1.SimpleFunc.integral f = MeasureTheory.L1.SimpleFunc.integral g
-- MeasureTheory.L1.SimpleFunc.integral_add
theorem integral_add (f g : α →₁ₛ[μ] E) : integral (f + g) = integral f + integral g 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (f g : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.integral (f + g) =
  MeasureTheory.L1.SimpleFunc.integral f + MeasureTheory.L1.SimpleFunc.integral g
-- MeasureTheory.L1.SimpleFunc.integral_smul
theorem integral_smul (c : 𝕜) (f : α →₁ₛ[μ] E) : integral (c • f) = c • integral f 
Arguments: {α : Type u_1} {E : Type u_2} {𝕜 : Type u_4} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace ℝ E] [SMulCommClass ℝ 𝕜 E] (c : 𝕜) (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.integral (c • f) = c • MeasureTheory.L1.SimpleFunc.integral f
-- MeasureTheory.L1.SimpleFunc.norm_integral_le_norm
theorem norm_integral_le_norm (f : α →₁ₛ[μ] E) : ‖integral f‖ ≤ ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] (f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ))
Goal: ‖MeasureTheory.L1.SimpleFunc.integral f‖ ≤ ‖f‖
-- MeasureTheory.L1.SimpleFunc.norm_Integral_le_one
theorem norm_Integral_le_one : ‖Integral‖ ≤ 1 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E]
Goal: ‖MeasureTheory.L1.SimpleFunc.integralCLM α E μ‖ ≤ 1
-- MeasureTheory.L1.SimpleFunc.posPart_toSimpleFunc
theorem posPart_toSimpleFunc (f : α →₁ₛ[μ] ℝ) :
    toSimpleFunc (posPart f) =ᵐ[μ] (toSimpleFunc f).posPart 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (MeasureTheory.L1.SimpleFunc.posPart f)) =ᵐ[μ]
  ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f).posPart
-- MeasureTheory.L1.SimpleFunc.negPart_toSimpleFunc
theorem negPart_toSimpleFunc (f : α →₁ₛ[μ] ℝ) :
    toSimpleFunc (negPart f) =ᵐ[μ] (toSimpleFunc f).negPart 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ))
Goal: ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc (MeasureTheory.L1.SimpleFunc.negPart f)) =ᵐ[μ]
  ↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f).negPart
-- MeasureTheory.L1.SimpleFunc.integral_eq_norm_posPart_sub
theorem integral_eq_norm_posPart_sub (f : α →₁ₛ[μ] ℝ) : integral f = ‖posPart f‖ - ‖negPart f‖ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ))
Goal: MeasureTheory.L1.SimpleFunc.integral f =
  ‖MeasureTheory.L1.SimpleFunc.posPart f‖ - ‖MeasureTheory.L1.SimpleFunc.negPart f‖
-- MeasureTheory.L1.integral_def
irreducible_def integral : (α →₁[μ] E) → E 
Arguments: {α : Type u_5} {E : Type u_6} [NormedAddCommGroup E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [NormedSpace ℝ E] [CompleteSpace E]
Goal: MeasureTheory.L1.integral = ⇑MeasureTheory.L1.integralCLM