ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.ConvolutionExists.distrib_add
theorem ConvolutionExists.distrib_add (hfg : ConvolutionExists f g L μ)
    (hfg' : ConvolutionExists f g' L μ) : f ⋆[L, μ] (g + g') = f ⋆[L, μ] g + 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' : 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] (hfg : MeasureTheory.ConvolutionExists f g L μ) (hfg' : MeasureTheory.ConvolutionExists f g' L μ)
Goal: MeasureTheory.convolution f (g + g') L μ = MeasureTheory.convolution f g L μ + MeasureTheory.convolution f g' L μ
-- MeasureTheory.ConvolutionExistsAt.add_distrib
theorem ConvolutionExistsAt.add_distrib {x : G} (hfg : ConvolutionExistsAt f g x L μ)
    (hfg' : ConvolutionExistsAt f' g x L μ) :
    ((f + f') ⋆[L, μ] 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 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] {x : G} (hfg : MeasureTheory.ConvolutionExistsAt f g x L μ) (hfg' : MeasureTheory.ConvolutionExistsAt f' g x L μ)
Goal: MeasureTheory.convolution (f + f') g L μ x = MeasureTheory.convolution f g L μ x + MeasureTheory.convolution f' g L μ x
-- MeasureTheory.ConvolutionExists.add_distrib
theorem ConvolutionExists.add_distrib (hfg : ConvolutionExists f g L μ)
    (hfg' : ConvolutionExists f' g L μ) : (f + f') ⋆[L, μ] g = f ⋆[L, μ] g + 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 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] (hfg : MeasureTheory.ConvolutionExists f g L μ) (hfg' : MeasureTheory.ConvolutionExists f' g L μ)
Goal: MeasureTheory.convolution (f + f') g L μ = MeasureTheory.convolution f g L μ + MeasureTheory.convolution f' g L μ
-- MeasureTheory.convolution_mono_right
theorem convolution_mono_right {f g g' : G → ℝ} (hfg : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ)
    (hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ) (hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) :
    (f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x 
Arguments: {G : Type uG} {x : G} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] {f g g' : G → ℝ} (hfg : MeasureTheory.ConvolutionExistsAt f g x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hfg' : MeasureTheory.ConvolutionExistsAt f g' x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hf : ∀ (x : G), 0 ≤ f x) (hg : ∀ (x : G), g x ≤ g' x)
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x ≤
  MeasureTheory.convolution f g' (ContinuousLinearMap.lsmul ℝ ℝ) μ x
-- MeasureTheory.convolution_mono_right_of_nonneg
theorem convolution_mono_right_of_nonneg {f g g' : G → ℝ}
    (hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ) (hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x)
    (hg' : ∀ x, 0 ≤ g' x) : (f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x 
Arguments: {G : Type uG} {x : G} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] {f g g' : G → ℝ} (hfg' : MeasureTheory.ConvolutionExistsAt f g' x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hf : ∀ (x : G), 0 ≤ f x) (hg : ∀ (x : G), g x ≤ g' x) (hg' : ∀ (x : G), 0 ≤ g' x)
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x ≤
  MeasureTheory.convolution f g' (ContinuousLinearMap.lsmul ℝ ℝ) μ x
-- MeasureTheory.convolution_congr
theorem convolution_congr [MeasurableAdd₂ G] [MeasurableNeg G] [SFinite μ]
    [IsAddRightInvariant μ] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g') : f ⋆[L, μ] g = 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 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] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g')
Goal: MeasureTheory.convolution f g L μ = MeasureTheory.convolution f' g' L μ
-- MeasureTheory.support_convolution_subset_swap
theorem support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f 
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]
Goal: Function.support (MeasureTheory.convolution f g L μ) ⊆ Function.support g + Function.support f
-- MeasureTheory.Integrable.integrable_convolution
theorem Integrable.integrable_convolution (hf : Integrable f μ)
    (hg : Integrable g μ) : Integrable (f ⋆[L, μ] g) 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} [NormedSpace ℝ F] [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ)
Goal: MeasureTheory.Integrable (MeasureTheory.convolution f g L μ) μ
-- HasCompactSupport.convolution
protected theorem _root_.HasCompactSupport.convolution [T2Space G] (hcf : HasCompactSupport f)
    (hcg : HasCompactSupport g) : HasCompactSupport (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] [TopologicalSpace G] [TopologicalAddGroup G] [T2Space G] (hcf : HasCompactSupport f) (hcg : HasCompactSupport g)
Goal: HasCompactSupport (MeasureTheory.convolution f g L μ)
-- MeasureTheory.continuousOn_convolution_right_with_param
theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
    (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
    (hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
    ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [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] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [TopologicalSpace P] {g : P → G → E'} {s : Set P} {k : Set G} (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ Set.univ))
Goal: ContinuousOn (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
-- MeasureTheory.continuousOn_convolution_right_with_param_comp
theorem continuousOn_convolution_right_with_param_comp {s : Set P} {v : P → G}
    (hv : ContinuousOn v s) {g : P → G → E'} {k : Set G} (hk : IsCompact k)
    (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
    (hg : ContinuousOn (↿g) (s ×ˢ univ)) : ContinuousOn (fun x => (f ⋆[L, μ] g x) (v x)) s 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [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] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [TopologicalSpace P] {s : Set P} {v : P → G} (hv : ContinuousOn v s) {g : P → G → E'} {k : Set G} (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ Set.univ))
Goal: ContinuousOn (fun x => MeasureTheory.convolution f (g x) L μ (v x)) s
-- HasCompactSupport.continuous_convolution_right
theorem _root_.HasCompactSupport.continuous_convolution_right (hcg : HasCompactSupport g)
    (hf : LocallyIntegrable f μ) (hg : Continuous g) : Continuous (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] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : Continuous g)
Goal: Continuous (MeasureTheory.convolution f g L μ)
-- BddAbove.continuous_convolution_right_of_integrable
theorem _root_.BddAbove.continuous_convolution_right_of_integrable
    [FirstCountableTopology G] [SecondCountableTopologyEither G E']
    (hbg : BddAbove (range fun x => ‖g x‖)) (hf : Integrable f μ) (hg : Continuous g) :
    Continuous (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] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [FirstCountableTopology G] [SecondCountableTopologyEither G E'] (hbg : BddAbove (Set.range fun x => ‖g x‖)) (hf : MeasureTheory.Integrable f μ) (hg : Continuous g)
Goal: Continuous (MeasureTheory.convolution f g L μ)
-- MeasureTheory.support_convolution_subset
theorem support_convolution_subset : support (f ⋆[L, μ] g) ⊆ support f + support 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] [AddCommGroup G]
Goal: Function.support (MeasureTheory.convolution f g L μ) ⊆ Function.support f + Function.support g
-- MeasureTheory.convolution_flip
theorem convolution_flip : g ⋆[L.flip, μ] f = 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] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G]
Goal: MeasureTheory.convolution g f L.flip μ = MeasureTheory.convolution f g L μ
-- MeasureTheory.convolution_eq_swap
theorem convolution_eq_swap : (f ⋆[L, μ] g) x = ∫ 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} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G]
Goal: MeasureTheory.convolution f g L μ x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ
-- MeasureTheory.convolution_lsmul_swap
theorem convolution_lsmul_swap {f : G → 𝕜} {g : G → F} :
    (f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f (x - t) • g 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] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] {f : G → 𝕜} {g : G → F}
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.lsmul 𝕜 𝕜) μ x = ∫ (t : G), f (x - t) • g t ∂μ
-- MeasureTheory.convolution_mul_swap
theorem convolution_mul_swap [NormedSpace ℝ 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
    (f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f (x - t) * g t ∂μ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {x : G} [NontriviallyNormedField 𝕜] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] [NormedSpace ℝ 𝕜] {f g : G → 𝕜}
Goal: MeasureTheory.convolution f g (ContinuousLinearMap.mul 𝕜 𝕜) μ x = ∫ (t : G), f (x - t) * g t ∂μ
-- MeasureTheory.convolution_neg_of_neg_eq
theorem convolution_neg_of_neg_eq (h1 : ∀ᵐ x ∂μ, f (-x) = f x) (h2 : ∀ᵐ x ∂μ, g (-x) = 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 → 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] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] (h1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x) (h2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x)
Goal: MeasureTheory.convolution f g L μ (-x) = MeasureTheory.convolution f g L μ x
-- HasCompactSupport.continuous_convolution_left
theorem _root_.HasCompactSupport.continuous_convolution_left
    (hcf : HasCompactSupport f) (hf : Continuous f) (hg : LocallyIntegrable g μ) :
    Continuous (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] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] (hcf : HasCompactSupport f) (hf : Continuous f) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: Continuous (MeasureTheory.convolution f g L μ)
-- BddAbove.continuous_convolution_left_of_integrable
theorem _root_.BddAbove.continuous_convolution_left_of_integrable
    [FirstCountableTopology G] [SecondCountableTopologyEither G E]
    (hbf : BddAbove (range fun x => ‖f x‖)) (hf : Continuous f) (hg : Integrable g μ) :
    Continuous (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] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [TopologicalSpace G] [TopologicalAddGroup G] [BorelSpace G] [FirstCountableTopology G] [SecondCountableTopologyEither G E] (hbf : BddAbove (Set.range fun x => ‖f x‖)) (hf : Continuous f) (hg : MeasureTheory.Integrable g μ)
Goal: Continuous (MeasureTheory.convolution f g L μ)
-- MeasureTheory.convolution_eq_right'
theorem convolution_eq_right' {x₀ : G} {R : ℝ} (hf : support f ⊆ ball (0 : G) R)
    (hg : ∀ x ∈ ball x₀ R, g x = g x₀) : (f ⋆[L, μ] g) x₀ = ∫ t, L (f t) (g x₀) ∂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} [NormedSpace ℝ F] [SeminormedAddCommGroup G] {x₀ : G} {R : ℝ} (hf : Function.support f ⊆ Metric.ball 0 R) (hg : ∀ x ∈ Metric.ball x₀ R, g x = g x₀)
Goal: MeasureTheory.convolution f g L μ x₀ = ∫ (t : G), (L (f t)) (g x₀) ∂μ
-- MeasureTheory.dist_convolution_le'
theorem dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : Integrable f μ)
    (hf : support f ⊆ ball (0 : G) R) (hmg : AEStronglyMeasurable g μ)
    (hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
    dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ (‖L‖ * ∫ x, ‖f x‖ ∂μ) * ε 
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] [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : MeasureTheory.Integrable f μ) (hf : Function.support f ⊆ Metric.ball 0 R) (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ R, dist (g x) z₀ ≤ ε)
Goal: dist (MeasureTheory.convolution f g L μ x₀) (∫ (t : G), (L (f t)) z₀ ∂μ) ≤ (‖L‖ * ∫ (x : G), ‖f x‖ ∂μ) * ε
-- MeasureTheory.dist_convolution_le
theorem dist_convolution_le {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε)
    (hf : support f ⊆ ball (0 : G) R) (hnf : ∀ x, 0 ≤ f x) (hintf : ∫ x, f x ∂μ = 1)
    (hmg : AEStronglyMeasurable g μ) (hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
    dist ((f ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) z₀ ≤ ε 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [NormedSpace ℝ E'] [CompleteSpace E'] {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hf : Function.support f ⊆ Metric.ball 0 R) (hnf : ∀ (x : G), 0 ≤ f x) (hintf : ∫ (x : G), f x ∂μ = 1) (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ R, dist (g x) z₀ ≤ ε)
Goal: dist (MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) z₀ ≤ ε
-- MeasureTheory.convolution_tendsto_right
theorem convolution_tendsto_right {ι} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'}
    {φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
    (hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1)
    -- todo: we could weaken this to "the integral tends to 1"
    (hφ : Tendsto (fun n => support (φ n)) l (𝓝 0).smallSets)
    (hmg : ∀ᶠ i in l, AEStronglyMeasurable (g i) μ) (hcg : Tendsto (uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀))
    (hk : Tendsto k l (𝓝 x₀)) :
    Tendsto (fun i : ι => (φ i ⋆[lsmul ℝ ℝ, μ] g i : G → E') (k i)) l (𝓝 z₀) 
Arguments: {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [NormedSpace ℝ E'] [CompleteSpace E'] {ι : Type u_1} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'} {φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x) (hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1) (hφ : Filter.Tendsto (fun n => Function.support (φ n)) l (𝓝 0).smallSets) (hmg : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (g i) μ) (hcg : Filter.Tendsto (Function.uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀)) (hk : Filter.Tendsto k l (𝓝 x₀))
Goal: Filter.Tendsto (fun i => MeasureTheory.convolution (φ i) (g i) (ContinuousLinearMap.lsmul ℝ ℝ) μ (k i)) l (𝓝 z₀)
-- MeasureTheory.integral_convolution
theorem integral_convolution [MeasurableAdd₂ G] [MeasurableNeg G] [NormedSpace ℝ E]
    [NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : Integrable f ν)
    (hg : Integrable g μ) : ∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂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'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G] [MeasurableNeg G] [NormedSpace ℝ E] [NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : MeasureTheory.Integrable f ν) (hg : MeasureTheory.Integrable g μ)
Goal: ∫ (x : G), MeasureTheory.convolution f g L ν x ∂μ = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
-- MeasureTheory.convolution_assoc'
theorem convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
    {x₀ : G} (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
    (hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt g k x L₄ μ)
    (hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)) :
    ((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] g ⋆[L₄, μ] k) x₀ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} {F' : Type uF'} {F'' : Type uF''} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup F'] [NormedSpace ℝ F'] [NormedSpace 𝕜 F'] [CompleteSpace F'] [NormedAddCommGroup F''] [NormedSpace ℝ F''] [NormedSpace 𝕜 F''] [CompleteSpace F''] {k : G → E''} (L₂ : F →L[𝕜] E'' →L[𝕜] F') (L₃ : E →L[𝕜] F'' →L[𝕜] F') (L₄ : E' →L[𝕜] E'' →L[𝕜] F'') [AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G] [ν.IsAddRightInvariant] [MeasurableNeg G] (hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)) {x₀ : G} (hfg : ∀ᵐ (y : G) ∂μ, MeasureTheory.ConvolutionExistsAt f g y L ν) (hgk : ∀ᵐ (x : G) ∂ν, MeasureTheory.ConvolutionExistsAt g k x L₄ μ) (hi : MeasureTheory.Integrable (Function.uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν))
Goal: MeasureTheory.convolution (MeasureTheory.convolution f g L ν) k L₂ μ x₀ =
  MeasureTheory.convolution f (MeasureTheory.convolution g k L₄ μ) L₃ ν x₀