ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Lp.mem_boundedContinuousFunction_iff
theorem MeasureTheory.Lp.mem_boundedContinuousFunction_iff {f : Lp E p μ} :
    f ∈ MeasureTheory.Lp.boundedContinuousFunction E p μ ↔
      ∃ f₀ : α →ᵇ E, f₀.toContinuousMap.toAEEqFun μ = (f : α →ₘ[μ] E) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] {f : ↥(MeasureTheory.Lp E p μ)}
Goal: f ∈ MeasureTheory.Lp.boundedContinuousFunction E p μ ↔ ∃ f₀, ContinuousMap.toAEEqFun μ f₀.toContinuousMap = ↑f
-- BoundedContinuousFunction.mem_Lp
theorem mem_Lp (f : α →ᵇ E) : f.toContinuousMap.toAEEqFun μ ∈ Lp E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] (f : α →ᵇ E)
Goal: ContinuousMap.toAEEqFun μ f.toContinuousMap ∈ MeasureTheory.Lp E p μ
-- BoundedContinuousFunction.Lp_nnnorm_le
theorem Lp_nnnorm_le (f : α →ᵇ E) :
    ‖(⟨f.toContinuousMap.toAEEqFun μ, mem_Lp f⟩ : Lp E p μ)‖₊ ≤
      measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖₊ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] (f : α →ᵇ E)
Goal: ‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖₊ ≤ MeasureTheory.measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖₊
-- BoundedContinuousFunction.Lp_norm_le
theorem Lp_norm_le (f : α →ᵇ E) :
    ‖(⟨f.toContinuousMap.toAEEqFun μ, mem_Lp f⟩ : Lp E p μ)‖ ≤
      measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] (f : α →ᵇ E)
Goal: ‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖ ≤ ↑(MeasureTheory.measureUnivNNReal μ) ^ p.toReal⁻¹ * ‖f‖
-- BoundedContinuousFunction.range_toLpHom
theorem range_toLpHom [Fact (1 ≤ p)] :
    ((toLpHom p μ).range : AddSubgroup (Lp E p μ)) =
      MeasureTheory.Lp.boundedContinuousFunction E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] [Fact (1 ≤ p)]
Goal: (BoundedContinuousFunction.toLpHom p μ).range = MeasureTheory.Lp.boundedContinuousFunction E p μ
-- BoundedContinuousFunction.coeFn_toLp
theorem coeFn_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] (f : α →ᵇ E) :
    toLp (E := E) p μ 𝕜 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] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] (𝕜 : Type u_5) [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E] (f : α →ᵇ E)
Goal: ↑↑((BoundedContinuousFunction.toLp p μ 𝕜) f) =ᵐ[μ] ⇑f
-- BoundedContinuousFunction.range_toLp
theorem range_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] :
    (LinearMap.range (toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ)).toAddSubgroup =
      MeasureTheory.Lp.boundedContinuousFunction E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (LinearMap.range (BoundedContinuousFunction.toLp p μ 𝕜)).toAddSubgroup =
  MeasureTheory.Lp.boundedContinuousFunction E p μ
-- BoundedContinuousFunction.toLp_norm_le
theorem toLp_norm_le [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] :
    ‖(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ)‖ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
Goal: ‖BoundedContinuousFunction.toLp p μ 𝕜‖ ≤ ↑(MeasureTheory.measureUnivNNReal μ) ^ p.toReal⁻¹
-- BoundedContinuousFunction.toLp_inj
theorem toLp_inj {f g : α →ᵇ E} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E] :
    toLp (E := E) p μ 𝕜 f = toLp (E := E) p μ 𝕜 g ↔ f = g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] {f g : α →ᵇ E} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (BoundedContinuousFunction.toLp p μ 𝕜) f = (BoundedContinuousFunction.toLp p μ 𝕜) g ↔ f = g
-- BoundedContinuousFunction.toLp_injective
theorem toLp_injective [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E] :
    Function.Injective (⇑(toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ)) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: Function.Injective ⇑(BoundedContinuousFunction.toLp p μ 𝕜)
-- ContinuousMap.range_toLp
theorem range_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] :
    (LinearMap.range (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ)).toAddSubgroup =
      MeasureTheory.Lp.boundedContinuousFunction E p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} (p : ℝ≥0∞) (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (LinearMap.range (ContinuousMap.toLp p μ 𝕜)).toAddSubgroup = MeasureTheory.Lp.boundedContinuousFunction E p μ
-- ContinuousMap.coeFn_toLp
theorem coeFn_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E)) :
    toLp (E := E) p μ 𝕜 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] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E))
Goal: ↑↑((ContinuousMap.toLp p μ 𝕜) f) =ᵐ[μ] ⇑f
-- ContinuousMap.toLp_def
theorem toLp_def [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E)) :
    toLp (E := E) p μ 𝕜 f =
      BoundedContinuousFunction.toLp (E := E) p μ 𝕜 (linearIsometryBoundedOfCompact α E 𝕜 f) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E))
Goal: (ContinuousMap.toLp p μ 𝕜) f =
  (BoundedContinuousFunction.toLp p μ 𝕜) ((ContinuousMap.linearIsometryBoundedOfCompact α E 𝕜) f)
-- ContinuousMap.toLp_comp_toContinuousMap
@[simp]
theorem toLp_comp_toContinuousMap [NormedField 𝕜] [NormedSpace 𝕜 E] (f : α →ᵇ E) :
    toLp (E := E) p μ 𝕜 f.toContinuousMap = BoundedContinuousFunction.toLp (E := E) p μ 𝕜 f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E] (f : α →ᵇ E)
Goal: (ContinuousMap.toLp p μ 𝕜) f.toContinuousMap = (BoundedContinuousFunction.toLp p μ 𝕜) f
-- ContinuousMap.coe_toLp
@[simp]
theorem coe_toLp [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E)) :
    (toLp (E := E) p μ 𝕜 f : α →ₘ[μ] E) = f.toAEEqFun μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NormedField 𝕜] [NormedSpace 𝕜 E] (f : C(α, E))
Goal: ↑((ContinuousMap.toLp p μ 𝕜) f) = ContinuousMap.toAEEqFun μ f
-- ContinuousMap.toLp_injective
theorem toLp_injective [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E] :
    Function.Injective (⇑(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ)) 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: Function.Injective ⇑(ContinuousMap.toLp p μ 𝕜)
-- ContinuousMap.toLp_inj
theorem toLp_inj {f g : C(α, E)} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E] :
    toLp (E := E) p μ 𝕜 f = toLp (E := E) p μ 𝕜 g ↔ f = g 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] {f g : C(α, E)} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
Goal: (ContinuousMap.toLp p μ 𝕜) f = (ContinuousMap.toLp p μ 𝕜) g ↔ f = g
-- ContinuousMap.hasSum_of_hasSum_Lp
theorem hasSum_of_hasSum_Lp {β : Type*} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
    {g : β → C(α, E)} {f : C(α, E)} (hg : Summable g)
    (hg2 : HasSum (toLp (E := E) p μ 𝕜 ∘ g) (toLp (E := E) p μ 𝕜 f)) : HasSum g f 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] {β : Type u_6} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E] {g : β → C(α, E)} {f : C(α, E)} (hg : Summable g) (hg2 : HasSum (⇑(ContinuousMap.toLp p μ 𝕜) ∘ g) ((ContinuousMap.toLp p μ 𝕜) f))
Goal: HasSum g f
-- ContinuousMap.toLp_norm_eq_toLp_norm_coe
theorem toLp_norm_eq_toLp_norm_coe :
    ‖(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ)‖ =
      ‖(BoundedContinuousFunction.toLp p μ 𝕜 : (α →ᵇ E) →L[𝕜] Lp E p μ)‖ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
Goal: ‖ContinuousMap.toLp p μ 𝕜‖ = ‖BoundedContinuousFunction.toLp p μ 𝕜‖
-- ContinuousMap.toLp_norm_le
theorem toLp_norm_le :
    ‖(toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ)‖ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) [NormedAddCommGroup E] [TopologicalSpace α] [BorelSpace α] [SecondCountableTopologyEither α E] [CompactSpace α] [MeasureTheory.IsFiniteMeasure μ] {𝕜 : Type u_5} [Fact (1 ≤ p)] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
Goal: ‖ContinuousMap.toLp p μ 𝕜‖ ≤ ↑(MeasureTheory.measureUnivNNReal μ) ^ p.toReal⁻¹
-- MeasureTheory.Lp.pow_mul_meas_ge_le_norm
theorem pow_mul_meas_ge_le_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
    (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖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)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ℝ≥0∞)
Goal: (ε * μ {x | ε ≤ ↑‖↑↑f x‖₊ ^ p.toReal}) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖f‖
-- MeasureTheory.Lp.mul_meas_ge_le_pow_norm
theorem mul_meas_ge_le_pow_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal } ≤ ENNReal.ofReal ‖f‖ ^ p.toReal 
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)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ℝ≥0∞)
Goal: ε * μ {x | ε ≤ ↑‖↑↑f x‖₊ ^ p.toReal} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal
-- MeasureTheory.Lp.mul_meas_ge_le_pow_norm'
theorem mul_meas_ge_le_pow_norm' (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (ε : ℝ≥0∞) : ε ^ p.toReal * μ { x | ε ≤ ‖f x‖₊ } ≤ ENNReal.ofReal ‖f‖ ^ p.toReal 
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)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (ε : ℝ≥0∞)
Goal: ε ^ p.toReal * μ {x | ε ≤ ↑‖↑↑f x‖₊} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal
-- MeasureTheory.Lp.meas_ge_le_mul_pow_norm
theorem meas_ge_le_mul_pow_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {ε : ℝ≥0∞}
    (hε : ε ≠ 0) : μ { x | ε ≤ ‖f x‖₊ } ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal 
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)) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: μ {x | ε ≤ ↑‖↑↑f x‖₊} ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal
-- MeasurableAdd.measurable_add_const
class MeasurableAdd (M : Type*) [MeasurableSpace M] [Add M] : Prop where
  measurable_const_add : ∀ c : M, Measurable (c + ·)
  measurable_add_const : ∀ c : M, Measurable (· + c)
Arguments: {M : Type u_2} [MeasurableSpace M] [Add M] [self : MeasurableAdd M] (c : M)
Goal: Measurable fun x => x + c
-- MeasurableAdd.measurable_const_add
class MeasurableAdd (M : Type*) [MeasurableSpace M] [Add M] : Prop where
  measurable_const_add : ∀ c : M, Measurable (c + ·)
  measurable_add_const : ∀ c : M, Measurable (· + c)
Arguments: {M : Type u_2} [MeasurableSpace M] [Add M] [self : MeasurableAdd M] (c : M)
Goal: Measurable fun x => c + x
-- MeasurableAdd₂.measurable_add
class MeasurableAdd₂ (M : Type*) [MeasurableSpace M] [Add M] : Prop where
  measurable_add : Measurable fun p : M × M => p.1 + p.2
Arguments: {M : Type u_2} [MeasurableSpace M] [Add M] [self : MeasurableAdd₂ M]
Goal: Measurable fun p => p.1 + p.2
-- MeasurableMul.measurable_mul_const
@[to_additive]
class MeasurableMul (M : Type*) [MeasurableSpace M] [Mul M] : Prop where
  measurable_const_mul : ∀ c : M, Measurable (c * ·)
  measurable_mul_const : ∀ c : M, Measurable (· * c)
Arguments: {M : Type u_2} [MeasurableSpace M] [Mul M] [self : MeasurableMul M] (c : M)
Goal: Measurable fun x => x * c
-- MeasurableMul.measurable_const_mul
@[to_additive]
class MeasurableMul (M : Type*) [MeasurableSpace M] [Mul M] : Prop where
  measurable_const_mul : ∀ c : M, Measurable (c * ·)
  measurable_mul_const : ∀ c : M, Measurable (· * c)
Arguments: {M : Type u_2} [MeasurableSpace M] [Mul M] [self : MeasurableMul M] (c : M)
Goal: Measurable fun x => c * x
-- MeasurableMul₂.measurable_mul
@[to_additive MeasurableAdd₂]
class MeasurableMul₂ (M : Type*) [MeasurableSpace M] [Mul M] : Prop where
  measurable_mul : Measurable fun p : M × M => p.1 * p.2
Arguments: {M : Type u_2} [MeasurableSpace M] [Mul M] [self : MeasurableMul₂ M]
Goal: Measurable fun p => p.1 * p.2
-- Measurable.const_mul
@[to_additive (attr := fun_prop, measurability)]
theorem Measurable.const_mul [MeasurableMul M] (hf : Measurable f) (c : M) :
    Measurable fun x => c * f x 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f : α → M} [MeasurableMul M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => c * f x
-- Measurable.const_add
@[to_additive (attr := fun_prop, measurability)]
theorem Measurable.const_mul [MeasurableMul M] (hf : Measurable f) (c : M) :
    Measurable fun x => c * f x 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f : α → M} [MeasurableAdd M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => c + f x
-- AEMeasurable.const_add
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_mul [MeasurableMul M] (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c * f x) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => c + f x) μ
-- AEMeasurable.const_mul
@[to_additive (attr := measurability)]
theorem AEMeasurable.const_mul [MeasurableMul M] (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => c * f x) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableMul M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => c * f x) μ
-- Measurable.add_const
@[to_additive (attr := measurability)]
theorem Measurable.mul_const [MeasurableMul M] (hf : Measurable f) (c : M) :
    Measurable fun x => f x * c 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f : α → M} [MeasurableAdd M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => f x + c
-- Measurable.mul_const
@[to_additive (attr := measurability)]
theorem Measurable.mul_const [MeasurableMul M] (hf : Measurable f) (c : M) :
    Measurable fun x => f x * c 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f : α → M} [MeasurableMul M] (hf : Measurable f) (c : M)
Goal: Measurable fun x => f x * c
-- AEMeasurable.add_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.mul_const [MeasurableMul M] (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x * c) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => f x + c) μ
-- AEMeasurable.mul_const
@[to_additive (attr := measurability)]
theorem AEMeasurable.mul_const [MeasurableMul M] (hf : AEMeasurable f μ) (c : M) :
    AEMeasurable (fun x => f x * c) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f : α → M} {μ : MeasureTheory.Measure α} [MeasurableMul M] (hf : AEMeasurable f μ) (c : M)
Goal: AEMeasurable (fun x => f x * c) μ
-- Measurable.add'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.mul' [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
    Measurable (f * g) 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f g : α → M} [MeasurableAdd₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f + g)
-- Measurable.mul'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.mul' [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
    Measurable (f * g) 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f g : α → M} [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable (f * g)
-- Measurable.mul
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.mul [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => f a * g a 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f g : α → M} [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a * g a
-- Measurable.add
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem Measurable.mul [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
    Measurable fun a => f a * g a 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f g : α → M} [MeasurableAdd₂ M] (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => f a + g a
-- AEMeasurable.add'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.mul' [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f * g) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f + g) μ
-- AEMeasurable.mul'
@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.mul' [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (f * g) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (f * g) μ
-- AEMeasurable.mul
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.mul [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a * g a) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a * g a) μ
-- AEMeasurable.add
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
theorem AEMeasurable.mul [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => f a * g a) μ 
Arguments: {M : Type u_2} {α : Type u_3} [MeasurableSpace M] [Add M] {m : MeasurableSpace α} {f g : α → M} {μ : MeasureTheory.Measure α} [MeasurableAdd₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => f a + g a) μ
-- measurable_sub_const'
@[to_additive " A version of `measurable_sub_const` that assumes `MeasurableAdd` instead of
  `MeasurableSub`. This can be nice to avoid unnecessary type-class assumptions. "]
theorem measurable_div_const' {G : Type*} [DivInvMonoid G] [MeasurableSpace G] [MeasurableMul G]
    (g : G) : Measurable fun h => h / g 
Arguments: {G : Type u_2} [SubNegMonoid G] [MeasurableSpace G] [MeasurableAdd G] (g : G)
Goal: Measurable fun h => h - g
-- measurable_div_const'
@[to_additive " A version of `measurable_sub_const` that assumes `MeasurableAdd` instead of
  `MeasurableSub`. This can be nice to avoid unnecessary type-class assumptions. "]
theorem measurable_div_const' {G : Type*} [DivInvMonoid G] [MeasurableSpace G] [MeasurableMul G]
    (g : G) : Measurable fun h => h / g 
Arguments: {G : Type u_2} [DivInvMonoid G] [MeasurableSpace G] [MeasurableMul G] (g : G)
Goal: Measurable fun h => h / g
-- MeasurablePow.measurable_pow
class MeasurablePow (β γ : Type*) [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] : Prop where
  measurable_pow : Measurable fun p : β × γ => p.1 ^ p.2
Arguments: {β : Type u_2} {γ : Type u_3} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [self : MeasurablePow β γ]
Goal: Measurable fun p => p.1 ^ p.2
-- Measurable.pow
@[aesop safe 20 apply (rule_sets := [Measurable])]
theorem Measurable.pow (hf : Measurable f) (hg : Measurable g) : Measurable fun x => f x ^ g x 
Arguments: {β : Type u_2} {γ : Type u_3} {α : Type u_4} [MeasurableSpace β] [MeasurableSpace γ] [Pow β γ] [MeasurablePow β γ] {m : MeasurableSpace α} {f : α → β} {g : α → γ} (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun x => f x ^ g x