ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Continuous.convolution_integrand_fst
theorem _root_.Continuous.convolution_integrand_fst [ContinuousSub G] (hg : Continuous g) (t : G) :
    Continuous fun x => L (f t) (g (x - t)) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [TopologicalSpace G] [ContinuousSub G] (hg : Continuous g) (t : G)
Goal: Continuous fun x => (L (f t)) (g (x - t))
-- HasCompactSupport.convolution_integrand_bound_left
theorem _root_.HasCompactSupport.convolution_integrand_bound_left (hcf : HasCompactSupport f)
    (hf : Continuous f) {x t : G} {s : Set G} (hx : x ∈ s) :
    ‖L (f (x - t)) (g t)‖ ≤
      (-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [TopologicalSpace G] (hcf : HasCompactSupport f) (hf : Continuous f) {x t : G} {s : Set G} (hx : x ∈ s)
Goal: ‖(L (f (x - t))) (g t)‖ ≤ (-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t
-- MeasureTheory.ConvolutionExistsAt.integrable
theorem ConvolutionExistsAt.integrable [Sub G] {x : G} (h : ConvolutionExistsAt f g x L μ) :
    Integrable (fun t => L (f t) (g (x - t))) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Sub G] {x : G} (h : MeasureTheory.ConvolutionExistsAt f g x L μ)
Goal: MeasureTheory.Integrable (fun t => (L (f t)) (g (x - t))) μ
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand'
theorem AEStronglyMeasurable.convolution_integrand' [MeasurableAdd₂ G]
    [MeasurableNeg G] [SFinite ν] (hf : AEStronglyMeasurable f ν)
    (hg : AEStronglyMeasurable g <| map (fun p : G × G => p.1 - p.2) (μ.prod ν)) :
    AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite ν] (hf : MeasureTheory.AEStronglyMeasurable f ν) (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map (fun p => p.1 - p.2) (μ.prod ν)))
Goal: MeasureTheory.AEStronglyMeasurable (fun p => (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd'
theorem AEStronglyMeasurable.convolution_integrand_snd'
    (hf : AEStronglyMeasurable f μ) {x : G}
    (hg : AEStronglyMeasurable g <| map (fun t => x - t) μ) :
    AEStronglyMeasurable (fun t => L (f t) (g (x - t))) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] (hf : MeasureTheory.AEStronglyMeasurable f μ) {x : G} (hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map (fun t => x - t) μ))
Goal: MeasureTheory.AEStronglyMeasurable (fun t => (L (f t)) (g (x - t))) μ
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd'
theorem AEStronglyMeasurable.convolution_integrand_swap_snd' {x : G}
    (hf : AEStronglyMeasurable f <| map (fun t => x - t) μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun t => L (f (x - t)) (g t)) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] {x : G} (hf : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun t => x - t) μ)) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun t => (L (f (x - t))) (g t)) μ
-- BddAbove.convolutionExistsAt'
theorem _root_.BddAbove.convolutionExistsAt' {x₀ : G} {s : Set G}
    (hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => -t + x₀) ⁻¹' s))) (hs : MeasurableSet s)
    (h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
    (hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) (μ.restrict s)) :
    ConvolutionExistsAt f g x₀ L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] {x₀ : G} {s : Set G} (hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => -t + x₀) ⁻¹' s))) (hs : MeasurableSet s) (h2s : (Function.support fun t => (L (f t)) (g (x₀ - t))) ⊆ s) (hf : MeasureTheory.IntegrableOn f s μ) (hmg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map (fun t => x₀ - t) (μ.restrict s)))
Goal: MeasureTheory.ConvolutionExistsAt f g x₀ L μ
-- MeasureTheory.ConvolutionExistsAt.ofNorm'
theorem ConvolutionExistsAt.ofNorm' {x₀ : G}
    (h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ)
    (hmf : AEStronglyMeasurable f μ) (hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) μ) :
    ConvolutionExistsAt f g x₀ L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd G] [MeasurableNeg G] {x₀ : G} (h : MeasureTheory.ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (ContinuousLinearMap.mul ℝ ℝ) μ) (hmf : MeasureTheory.AEStronglyMeasurable f μ) (hmg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map (fun t => x₀ - t) μ))
Goal: MeasureTheory.ConvolutionExistsAt f g x₀ L μ
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd
theorem AEStronglyMeasurable.convolution_integrand_snd (hf : AEStronglyMeasurable f μ)
    (hg : AEStronglyMeasurable g μ) (x : G) :
    AEStronglyMeasurable (fun t => L (f t) (g (x - t))) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (x : G)
Goal: MeasureTheory.AEStronglyMeasurable (fun t => (L (f t)) (g (x - t))) μ
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd
theorem AEStronglyMeasurable.convolution_integrand_swap_snd
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (x : G) :
    AEStronglyMeasurable (fun t => L (f (x - t)) (g t)) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (x : G)
Goal: MeasureTheory.AEStronglyMeasurable (fun t => (L (f (x - t))) (g t)) μ
-- MeasureTheory.ConvolutionExistsAt.ofNorm
theorem ConvolutionExistsAt.ofNorm {x₀ : G}
    (h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ)
    (hmf : AEStronglyMeasurable f μ) (hmg : AEStronglyMeasurable g μ) :
    ConvolutionExistsAt f g x₀ L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] {x₀ : G} (h : MeasureTheory.ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (ContinuousLinearMap.mul ℝ ℝ) μ) (hmf : MeasureTheory.AEStronglyMeasurable f μ) (hmg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.ConvolutionExistsAt f g x₀ L μ
-- MeasureTheory.AEStronglyMeasurable.convolution_integrand
theorem AEStronglyMeasurable.convolution_integrand (hf : AEStronglyMeasurable f ν)
    (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] [MeasureTheory.SFinite ν] (hf : MeasureTheory.AEStronglyMeasurable f ν) (hg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun p => (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
-- MeasureTheory.Integrable.convolution_integrand
theorem Integrable.convolution_integrand (hf : Integrable f ν) (hg : Integrable g μ) :
    Integrable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] [MeasureTheory.SFinite ν] (hf : MeasureTheory.Integrable f ν) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable (fun p => (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
-- MeasureTheory.Integrable.ae_convolution_exists
theorem Integrable.ae_convolution_exists (hf : Integrable f ν) (hg : Integrable g μ) :
    ∀ᵐ x ∂μ, ConvolutionExistsAt f g x L ν 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] [MeasureTheory.SFinite ν] (hf : MeasureTheory.Integrable f ν) (hg : MeasureTheory.Integrable g μ)
Goal: ∀ᵐ (x : G) ∂μ, MeasureTheory.ConvolutionExistsAt f g x L ν
-- HasCompactSupport.convolutionExistsAt
theorem _root_.HasCompactSupport.convolutionExistsAt {x₀ : G}
    (h : HasCompactSupport fun t => L (f t) (g (x₀ - t))) (hf : LocallyIntegrable f μ)
    (hg : Continuous g) : ConvolutionExistsAt f g x₀ L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] {x₀ : G} (h : HasCompactSupport fun t => (L (f t)) (g (x₀ - t))) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : Continuous g)
Goal: MeasureTheory.ConvolutionExistsAt f g x₀ L μ
-- HasCompactSupport.convolutionExists_right
theorem _root_.HasCompactSupport.convolutionExists_right (hcg : HasCompactSupport g)
    (hf : LocallyIntegrable f μ) (hg : Continuous g) : ConvolutionExists f g L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : Continuous g)
Goal: MeasureTheory.ConvolutionExists f g L μ
-- HasCompactSupport.convolutionExists_left_of_continuous_right
theorem _root_.HasCompactSupport.convolutionExists_left_of_continuous_right
    (hcf : HasCompactSupport f) (hf : LocallyIntegrable f μ) (hg : Continuous g) :
    ConvolutionExists f g L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (hcf : HasCompactSupport f) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : Continuous g)
Goal: MeasureTheory.ConvolutionExists f g L μ
-- BddAbove.convolutionExistsAt
theorem _root_.BddAbove.convolutionExistsAt [MeasurableAdd₂ G] [SFinite μ] {x₀ : G} {s : Set G}
    (hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => x₀ - t) ⁻¹' s))) (hs : MeasurableSet s)
    (h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
    (hmg : AEStronglyMeasurable g μ) : ConvolutionExistsAt f g x₀ L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [MeasurableNeg G] [μ.IsAddLeftInvariant] [MeasurableAdd₂ G] [MeasureTheory.SFinite μ] {x₀ : G} {s : Set G} (hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => x₀ - t) ⁻¹' s))) (hs : MeasurableSet s) (h2s : (Function.support fun t => (L (f t)) (g (x₀ - t))) ⊆ s) (hf : MeasureTheory.IntegrableOn f s μ) (hmg : MeasureTheory.AEStronglyMeasurable g μ)
Goal: MeasureTheory.ConvolutionExistsAt f g x₀ L μ
-- MeasureTheory.convolutionExistsAt_flip
theorem convolutionExistsAt_flip :
    ConvolutionExistsAt g f x L.flip μ ↔ ConvolutionExistsAt f g x L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [MeasurableNeg G] [μ.IsAddLeftInvariant] [MeasurableAdd G] [μ.IsNegInvariant]
Goal: MeasureTheory.ConvolutionExistsAt g f x L.flip μ ↔ MeasureTheory.ConvolutionExistsAt f g x L μ
-- MeasureTheory.ConvolutionExistsAt.integrable_swap
theorem ConvolutionExistsAt.integrable_swap (h : ConvolutionExistsAt f g x L μ) :
    Integrable (fun t => L (f (x - t)) (g t)) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [MeasurableNeg G] [μ.IsAddLeftInvariant] [MeasurableAdd G] [μ.IsNegInvariant] (h : MeasureTheory.ConvolutionExistsAt f g x L μ)
Goal: MeasureTheory.Integrable (fun t => (L (f (x - t))) (g t)) μ
-- MeasureTheory.convolutionExistsAt_iff_integrable_swap
theorem convolutionExistsAt_iff_integrable_swap :
    ConvolutionExistsAt f g x L μ ↔ Integrable (fun t => L (f (x - t)) (g t)) μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [MeasurableNeg G] [μ.IsAddLeftInvariant] [MeasurableAdd G] [μ.IsNegInvariant]
Goal: MeasureTheory.ConvolutionExistsAt f g x L μ ↔ MeasureTheory.Integrable (fun t => (L (f (x - t))) (g t)) μ
-- HasCompactSupport.convolutionExistsLeft
theorem _root_.HasCompactSupport.convolutionExistsLeft
    (hcf : HasCompactSupport f) (hf : Continuous f)
    (hg : LocallyIntegrable g μ) : ConvolutionExists f g L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (hcf : HasCompactSupport f) (hf : Continuous f) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: MeasureTheory.ConvolutionExists f g L μ
-- HasCompactSupport.convolutionExistsRightOfContinuousLeft
theorem _root_.HasCompactSupport.convolutionExistsRightOfContinuousLeft (hcg : HasCompactSupport g)
    (hf : Continuous f) (hg : LocallyIntegrable g μ) : ConvolutionExists f g L μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (hcg : HasCompactSupport g) (hf : Continuous f) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: MeasureTheory.ConvolutionExists f g L μ
-- MeasureTheory.convolution_def
theorem convolution_def [Sub G] : (f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [Sub G]
Goal: MeasureTheory.convolution f g L μ x = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ
-- MeasureTheory.convolution_lsmul
theorem convolution_lsmul [Sub G] {f : G → 𝕜} {g : G → F} :
    (f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f t • g (x - t) ∂μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {F : Type uF} [NormedAddCommGroup F] {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [Sub G] {f : G → 𝕜} {g : G → F}
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.lsmul 𝕜 𝕜) μ x = ∫ (t : G), f t • g (x - t) ∂μ
-- MeasureTheory.convolution_mul
theorem convolution_mul [Sub G] [NormedSpace ℝ 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
    (f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f t * g (x - t) ∂μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {x : G} [NontriviallyNormedField 𝕜] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Sub G] [NormedSpace ℝ 𝕜] {f g : G → 𝕜}
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.mul 𝕜 𝕜) μ x = ∫ (t : G), f t * g (x - t) ∂μ
-- MeasureTheory.smul_convolution
theorem smul_convolution [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : y • f ⋆[L, μ] g = y • (f ⋆[L, μ] g) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [SMulCommClass ℝ 𝕜 F] {y : 𝕜}
Goal: MeasureTheory.convolution (y • f) g L μ = y • MeasureTheory.convolution f g L μ
-- MeasureTheory.convolution_smul
theorem convolution_smul [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : f ⋆[L, μ] y • g = y • (f ⋆[L, μ] g) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [SMulCommClass ℝ 𝕜 F] {y : 𝕜}
Goal: MeasureTheory.convolution f (y • g) L μ = y • MeasureTheory.convolution f g L μ
-- MeasureTheory.zero_convolution
@[simp]
theorem zero_convolution : 0 ⋆[L, μ] g = 0 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G]
Goal: MeasureTheory.convolution 0 g L μ = 0
-- MeasureTheory.convolution_zero
@[simp]
theorem convolution_zero : f ⋆[L, μ] 0 = 0 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G]
Goal: MeasureTheory.convolution f 0 L μ = 0
-- MeasureTheory.ConvolutionExistsAt.distrib_add
theorem ConvolutionExistsAt.distrib_add {x : G} (hfg : ConvolutionExistsAt f g x L μ)
    (hfg' : ConvolutionExistsAt f g' x L μ) :
    (f ⋆[L, μ] (g + g')) x = (f ⋆[L, μ] g) x + (f ⋆[L, μ] g') x 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g g' : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] {x : G} (hfg : MeasureTheory.ConvolutionExistsAt f g x L μ) (hfg' : MeasureTheory.ConvolutionExistsAt f g' x L μ)
Goal: MeasureTheory.convolution f (g + g') L μ x = MeasureTheory.convolution f g L μ x + MeasureTheory.convolution f g' L μ x