ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- 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}
    (hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) (hk : AEStronglyMeasurable k μ)
    (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
    (hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ)
    (hfgk :
      ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀
        (mul ℝ ℝ) 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} (hf : MeasureTheory.AEStronglyMeasurable f ν) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hk : MeasureTheory.AEStronglyMeasurable k μ) (hfg : ∀ᵐ (y : G) ∂μ, MeasureTheory.ConvolutionExistsAt f g y L ν) (hgk :
  ∀ᵐ (x : G) ∂ν, MeasureTheory.ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (ContinuousLinearMap.mul ℝ ℝ) μ) (hfgk :
  MeasureTheory.ConvolutionExistsAt (fun x => ‖f x‖)
    (MeasureTheory.convolution (fun x => ‖g x‖) (fun x => ‖k x‖) (ContinuousLinearMap.mul ℝ ℝ) μ) x₀
    (ContinuousLinearMap.mul ℝ ℝ) 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
-- MeasureTheory.convolution_precompR_apply
theorem convolution_precompR_apply {g : G → E'' →L[𝕜] E'} (hf : LocallyIntegrable f μ)
    (hcg : HasCompactSupport g) (hg : Continuous g) (x₀ : G) (x : E'') :
    (f ⋆[L.precompR E'', μ] g) x₀ x = (f ⋆[L, μ] fun a => g a x) x₀ 
Arguments: {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f : G → E} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] {g : G → E'' →L[𝕜] E'} (hf : MeasureTheory.LocallyIntegrable f μ) (hcg : HasCompactSupport g) (hg : Continuous g) (x₀ : G) (x : E'')
Goal: (MeasureTheory.convolution f g (ContinuousLinearMap.precompR E'' L) μ x₀) x =
  MeasureTheory.convolution f (fun a => (g a) x) L μ x₀
-- HasCompactSupport.hasFDerivAt_convolution_right
theorem _root_.HasCompactSupport.hasFDerivAt_convolution_right (hcg : HasCompactSupport g)
    (hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G) :
    HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ 
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] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant] (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G)
Goal: HasFDerivAt (MeasureTheory.convolution f g L μ)
  (MeasureTheory.convolution f (fderiv 𝕜 g) (ContinuousLinearMap.precompR G L) μ x₀) x₀
-- HasCompactSupport.hasFDerivAt_convolution_left
theorem _root_.HasCompactSupport.hasFDerivAt_convolution_left [IsNegInvariant μ]
    (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 1 f) (hg : LocallyIntegrable g μ) (x₀ : G) :
    HasFDerivAt (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[L.precompL G, μ] g) x₀) x₀ 
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] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 1 f) (hg : MeasureTheory.LocallyIntegrable g μ) (x₀ : G)
Goal: HasFDerivAt (MeasureTheory.convolution f g L μ)
  (MeasureTheory.convolution (fderiv 𝕜 f) g (ContinuousLinearMap.precompL G L) μ x₀) x₀
-- HasCompactSupport.hasDerivAt_convolution_right
theorem _root_.HasCompactSupport.hasDerivAt_convolution_right (hf : LocallyIntegrable f₀ μ)
    (hcg : HasCompactSupport g₀) (hg : ContDiff 𝕜 1 g₀) (x₀ : 𝕜) :
    HasDerivAt (f₀ ⋆[L, μ] g₀) ((f₀ ⋆[L, μ] deriv g₀) x₀) x₀ 
Arguments: {𝕜 : Type u𝕜} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'} (L : E →L[𝕜] E' →L[𝕜] F) {μ : MeasureTheory.Measure 𝕜} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] (hf : MeasureTheory.LocallyIntegrable f₀ μ) (hcg : HasCompactSupport g₀) (hg : ContDiff 𝕜 1 g₀) (x₀ : 𝕜)
Goal: HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution f₀ (deriv g₀) L μ x₀) x₀
-- HasCompactSupport.hasDerivAt_convolution_left
theorem _root_.HasCompactSupport.hasDerivAt_convolution_left [IsNegInvariant μ]
    (hcf : HasCompactSupport f₀) (hf : ContDiff 𝕜 1 f₀) (hg : LocallyIntegrable g₀ μ) (x₀ : 𝕜) :
    HasDerivAt (f₀ ⋆[L, μ] g₀) ((deriv f₀ ⋆[L, μ] g₀) x₀) x₀ 
Arguments: {𝕜 : Type u𝕜} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'} (L : E →L[𝕜] E' →L[𝕜] F) {μ : MeasureTheory.Measure 𝕜} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [μ.IsNegInvariant] (hcf : HasCompactSupport f₀) (hf : ContDiff 𝕜 1 f₀) (hg : MeasureTheory.LocallyIntegrable g₀ μ) (x₀ : 𝕜)
Goal: HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution (deriv f₀) g₀ L μ x₀) x₀
-- MeasureTheory.hasFDerivAt_convolution_right_with_param
theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
    (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
    (hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)) (q₀ : P × G)
    (hq₀ : q₀.1 ∈ s) :
    HasFDerivAt (fun q : P × G => (f ⋆[L, μ] g q.1) q.2)
      ((f ⋆[L.precompR (P × G), μ] fun x : G => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀ 
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} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ Set.univ)) (q₀ : P × G) (hq₀ : q₀.1 ∈ s)
Goal: HasFDerivAt (fun q => MeasureTheory.convolution f (g q.1) L μ q.2)
  (MeasureTheory.convolution f (fun x => fderiv 𝕜 (↿g) (q₀.1, x)) (ContinuousLinearMap.precompR (P × G) L) μ q₀.2) q₀
-- MeasureTheory.contDiffOn_convolution_right_with_param_aux
theorem contDiffOn_convolution_right_with_param_aux {G : Type uP} {E' : Type uP} {F : Type uP}
    {P : Type uP} [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace 𝕜 E']
    [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G]
    {μ : Measure G}
    [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P]
    {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G}
    (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
    (hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
    ContDiffOn 𝕜 n (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) 
Arguments: {𝕜 : Type u𝕜} {E : Type uE} [NormedAddCommGroup E] [RCLike 𝕜] [NormedSpace 𝕜 E] {G E' F P : Type uP} [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ))
Goal: ContDiffOn 𝕜 n (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
-- MeasureTheory.contDiffOn_convolution_right_with_param
theorem contDiffOn_convolution_right_with_param {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
    {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k)
    (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
    (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
    ContDiffOn 𝕜 n (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] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ))
Goal: ContDiffOn 𝕜 n (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
-- MeasureTheory.contDiffOn_convolution_right_with_param_comp
theorem contDiffOn_convolution_right_with_param_comp {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {s : Set P}
    {v : P → G} (hv : ContDiffOn 𝕜 n v s) {f : G → E} {g : P → G → E'} {k : Set G} (hs : IsOpen s)
    (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
    (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) : ContDiffOn 𝕜 n (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] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {s : Set P} {v : P → G} (hv : ContDiffOn 𝕜 n v s) {f : G → E} {g : P → G → E'} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ))
Goal: ContDiffOn 𝕜 n (fun x => MeasureTheory.convolution f (g x) L μ (v x)) s
-- MeasureTheory.contDiffOn_convolution_left_with_param
theorem contDiffOn_convolution_left_with_param [μ.IsAddLeftInvariant] [μ.IsNegInvariant]
    (L : E' →L[𝕜] E →L[𝕜] F) {f : G → E} {n : ℕ∞} {g : P → G → E'} {s : Set P} {k : Set G}
    (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
    (hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
    ContDiffOn 𝕜 n (fun q : P × G => (g q.1 ⋆[L, μ] f) 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] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (L : E' →L[𝕜] E →L[𝕜] F) {f : G → E} {n : ℕ∞} {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ))
Goal: ContDiffOn 𝕜 n (fun q => MeasureTheory.convolution (g q.1) f L μ q.2) (s ×ˢ Set.univ)
-- MeasureTheory.contDiffOn_convolution_left_with_param_comp
theorem contDiffOn_convolution_left_with_param_comp [μ.IsAddLeftInvariant] [μ.IsNegInvariant]
    (L : E' →L[𝕜] E →L[𝕜] F) {s : Set P} {n : ℕ∞} {v : P → G} (hv : ContDiffOn 𝕜 n v s) {f : G → E}
    {g : P → G → E'} {k : Set G} (hs : IsOpen s) (hk : IsCompact k)
    (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
    (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) : ContDiffOn 𝕜 n (fun x => (g x ⋆[L, μ] f) (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] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (L : E' →L[𝕜] E →L[𝕜] F) {s : Set P} {n : ℕ∞} {v : P → G} (hv : ContDiffOn 𝕜 n v s) {f : G → E} {g : P → G → E'} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ))
Goal: ContDiffOn 𝕜 n (fun x => MeasureTheory.convolution (g x) f L μ (v x)) s
-- HasCompactSupport.contDiff_convolution_right
theorem _root_.HasCompactSupport.contDiff_convolution_right {n : ℕ∞} (hcg : HasCompactSupport g)
    (hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n (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'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) {n : ℕ∞} (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiff 𝕜 n g)
Goal: ContDiff 𝕜 n (MeasureTheory.convolution f g L μ)
-- HasCompactSupport.contDiff_convolution_left
theorem _root_.HasCompactSupport.contDiff_convolution_left [μ.IsAddLeftInvariant] [μ.IsNegInvariant]
    {n : ℕ∞} (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 n f) (hg : LocallyIntegrable g μ) :
    ContDiff 𝕜 n (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'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [CompleteSpace F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [μ.IsAddLeftInvariant] [μ.IsNegInvariant] {n : ℕ∞} (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 n f) (hg : MeasureTheory.LocallyIntegrable g μ)
Goal: ContDiff 𝕜 n (MeasureTheory.convolution f g L μ)
-- MeasureTheory.posConvolution_eq_convolution_indicator
theorem posConvolution_eq_convolution_indicator (f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F)
    (ν : Measure ℝ := by volume_tac) [NoAtoms ν] :
    posConvolution f g L ν = convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L ν 
Arguments: {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ E'] [NormedSpace ℝ F] (f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : MeasureTheory.Measure ℝ := by volume_tac) [MeasureTheory.NoAtoms ν]
Goal: MeasureTheory.posConvolution f g L ν = MeasureTheory.convolution ((Set.Ioi 0).indicator f) ((Set.Ioi 0).indicator g) L ν
-- MeasureTheory.integrable_posConvolution
theorem integrable_posConvolution {f : ℝ → E} {g : ℝ → E'} {μ ν : Measure ℝ} [SFinite μ]
    [SFinite ν] [IsAddRightInvariant μ] [NoAtoms ν] (hf : IntegrableOn f (Ioi 0) ν)
    (hg : IntegrableOn g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
    Integrable (posConvolution f g L ν) μ 
Arguments: {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ E'] [NormedSpace ℝ F] {f : ℝ → E} {g : ℝ → E'} {μ ν : MeasureTheory.Measure ℝ} [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasureTheory.NoAtoms ν] (hf : MeasureTheory.IntegrableOn f (Set.Ioi 0) ν) (hg : MeasureTheory.IntegrableOn g (Set.Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F)
Goal: MeasureTheory.Integrable (MeasureTheory.posConvolution f g L ν) μ
-- MeasureTheory.integral_posConvolution
theorem integral_posConvolution [CompleteSpace E] [CompleteSpace E'] {μ ν : Measure ℝ}
    [SFinite μ] [SFinite ν] [IsAddRightInvariant μ] [NoAtoms ν] {f : ℝ → E} {g : ℝ → E'}
    (hf : IntegrableOn f (Ioi 0) ν) (hg : IntegrableOn g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
    ∫ x : ℝ in Ioi 0, ∫ t : ℝ in (0)..x, L (f t) (g (x - t)) ∂ν ∂μ =
      L (∫ x : ℝ in Ioi 0, f x ∂ν) (∫ x : ℝ in Ioi 0, g x ∂μ) 
Arguments: {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ E'] [NormedSpace ℝ F] [CompleteSpace F] [CompleteSpace E] [CompleteSpace E'] {μ ν : MeasureTheory.Measure ℝ} [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasureTheory.NoAtoms ν] {f : ℝ → E} {g : ℝ → E'} (hf : MeasureTheory.IntegrableOn f (Set.Ioi 0) ν) (hg : MeasureTheory.IntegrableOn g (Set.Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F)
Goal: ∫ (x : ℝ) in Set.Ioi 0, ∫ (t : ℝ) in 0 ..x, (L (f t)) (g (x - t)) ∂ν ∂μ =
  (L (∫ (x : ℝ) in Set.Ioi 0, f x ∂ν)) (∫ (x : ℝ) in Set.Ioi 0, g x ∂μ)
-- IsLocalExtrOn.range_ne_top_of_hasStrictFDerivAt
theorem IsLocalExtrOn.range_ne_top_of_hasStrictFDerivAt
    (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀)
    (hφ' : HasStrictFDerivAt φ φ' x₀) : LinearMap.range (f'.prod φ') ≠ ⊤ 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {f : E → F} {φ : E → ℝ} {x₀ : E} {f' : E →L[ℝ] F} {φ' : E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀) (hφ' : HasStrictFDerivAt φ φ' x₀)
Goal: LinearMap.range (f'.prod φ') ≠ ⊤
-- IsLocalExtrOn.exists_linear_map_of_hasStrictFDerivAt
theorem IsLocalExtrOn.exists_linear_map_of_hasStrictFDerivAt
    (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀)
    (hφ' : HasStrictFDerivAt φ φ' x₀) :
    ∃ (Λ : Module.Dual ℝ F) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ ∀ x, Λ (f' x) + Λ₀ • φ' x = 0 
Arguments: {E : Type u_1} {F : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {f : E → F} {φ : E → ℝ} {x₀ : E} {f' : E →L[ℝ] F} {φ' : E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀) (hφ' : HasStrictFDerivAt φ φ' x₀)
Goal: ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∀ (x : E), Λ (f' x) + Λ₀ • φ' x = 0
-- IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d
theorem IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt_1d {f : E → ℝ} {f' : E →L[ℝ] ℝ}
    (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀)
    (hφ' : HasStrictFDerivAt φ φ' x₀) : ∃ a b : ℝ, (a, b) ≠ 0 ∧ a • f' + b • φ' = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {φ : E → ℝ} {x₀ : E} {φ' : E →L[ℝ] ℝ} {f : E → ℝ} {f' : E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀) (hφ' : HasStrictFDerivAt φ φ' x₀)
Goal: ∃ a b, (a, b) ≠ 0 ∧ a • f' + b • φ' = 0
-- IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt
theorem IsLocalExtrOn.exists_multipliers_of_hasStrictFDerivAt {ι : Type*} [Fintype ι]
    {f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ i, f i x = f i x₀} x₀)
    (hf' : ∀ i, HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀) :
    ∃ (Λ : ι → ℝ) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ (∑ i, Λ i • f' i) + Λ₀ • φ' = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {φ : E → ℝ} {x₀ : E} {φ' : E →L[ℝ] ℝ} {ι : Type u_3} [Fintype ι] {f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀) (hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀)
Goal: ∃ Λ Λ₀, (Λ, Λ₀) ≠ 0 ∧ ∑ i : ι, Λ i • f' i + Λ₀ • φ' = 0
-- IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt
theorem IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt {ι : Type*} [Finite ι] {f : ι → E → ℝ}
    {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ i, f i x = f i x₀} x₀)
    (hf' : ∀ i, HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀) :
    ¬LinearIndependent ℝ (Option.elim' φ' f' : Option ι → E →L[ℝ] ℝ) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {φ : E → ℝ} {x₀ : E} {φ' : E →L[ℝ] ℝ} {ι : Type u_3} [Finite ι] {f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀) (hf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀)
Goal: ¬LinearIndependent ℝ (Option.elim' φ' f')
-- hasDerivAt_update
theorem hasDerivAt_update (x : ι → 𝕜) (i : ι) (y : 𝕜) :
    HasDerivAt (Function.update x i) (Pi.single i (1 : 𝕜)) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] (x : ι → 𝕜) (i : ι) (y : 𝕜)
Goal: HasDerivAt (Function.update x i) (Pi.single i 1) y
-- hasDerivAt_single
theorem hasDerivAt_single (i : ι) (y : 𝕜) :
    HasDerivAt (Pi.single (f := fun _ ↦ 𝕜) i) (Pi.single i (1 : 𝕜)) y 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] (i : ι) (y : 𝕜)
Goal: HasDerivAt (Pi.single i) (Pi.single i 1) y
-- deriv_update
theorem deriv_update (x : ι → 𝕜) (i : ι) (y : 𝕜) :
    deriv (Function.update x i) y = Pi.single i (1 : 𝕜) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] (x : ι → 𝕜) (i : ι) (y : 𝕜)
Goal: deriv (Function.update x i) y = Pi.single i 1
-- deriv_single
theorem deriv_single (i : ι) (y : 𝕜) :
    deriv (Pi.single (f := fun _ ↦ 𝕜) i) y = Pi.single i (1 : 𝕜) 
Arguments: {𝕜 : Type u_1} {ι : Type u_2} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜] (i : ι) (y : 𝕜)
Goal: deriv (Pi.single i) y = Pi.single i 1
-- hasStrictDerivAt_zpow
theorem hasStrictDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
    HasStrictDerivAt (fun x => x ^ m) ((m : 𝕜) * x ^ (m - 1)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m)
Goal: HasStrictDerivAt (fun x => x ^ m) (↑m * x ^ (m - 1)) x