ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.differentiableOn
@[fun_prop]
protected theorem ContinuousLinearMap.differentiableOn : DifferentiableOn 𝕜 e s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (e : E →L[𝕜] F) {s : Set E}
Goal: DifferentiableOn 𝕜 (⇑e) s
-- IsBoundedLinearMap.hasFDerivAtFilter
theorem IsBoundedLinearMap.hasFDerivAtFilter (h : IsBoundedLinearMap 𝕜 f) :
    HasFDerivAtFilter f h.toContinuousLinearMap x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {L : Filter E} (h : IsBoundedLinearMap 𝕜 f)
Goal: HasFDerivAtFilter f h.toContinuousLinearMap x L
-- IsBoundedLinearMap.hasFDerivWithinAt
@[fun_prop]
theorem IsBoundedLinearMap.hasFDerivWithinAt (h : IsBoundedLinearMap 𝕜 f) :
    HasFDerivWithinAt f h.toContinuousLinearMap s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : IsBoundedLinearMap 𝕜 f)
Goal: HasFDerivWithinAt f h.toContinuousLinearMap s x
-- IsBoundedLinearMap.hasFDerivAt
@[fun_prop]
theorem IsBoundedLinearMap.hasFDerivAt (h : IsBoundedLinearMap 𝕜 f) :
    HasFDerivAt f h.toContinuousLinearMap x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : IsBoundedLinearMap 𝕜 f)
Goal: HasFDerivAt f h.toContinuousLinearMap x
-- IsBoundedLinearMap.differentiableAt
@[fun_prop]
theorem IsBoundedLinearMap.differentiableAt (h : IsBoundedLinearMap 𝕜 f) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : IsBoundedLinearMap 𝕜 f)
Goal: DifferentiableAt 𝕜 f x
-- IsBoundedLinearMap.differentiableWithinAt
@[fun_prop]
theorem IsBoundedLinearMap.differentiableWithinAt (h : IsBoundedLinearMap 𝕜 f) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : IsBoundedLinearMap 𝕜 f)
Goal: DifferentiableWithinAt 𝕜 f s x
-- IsBoundedLinearMap.fderiv
theorem IsBoundedLinearMap.fderiv (h : IsBoundedLinearMap 𝕜 f) :
    fderiv 𝕜 f x = h.toContinuousLinearMap 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : IsBoundedLinearMap 𝕜 f)
Goal: fderiv 𝕜 f x = h.toContinuousLinearMap
-- IsBoundedLinearMap.fderivWithin
theorem IsBoundedLinearMap.fderivWithin (h : IsBoundedLinearMap 𝕜 f)
    (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 f s x = h.toContinuousLinearMap 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E} (h : IsBoundedLinearMap 𝕜 f) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 f s x = h.toContinuousLinearMap
-- IsBoundedLinearMap.differentiable
@[fun_prop]
theorem IsBoundedLinearMap.differentiable (h : IsBoundedLinearMap 𝕜 f) : Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (h : IsBoundedLinearMap 𝕜 f)
Goal: Differentiable 𝕜 f
-- IsBoundedLinearMap.differentiableOn
@[fun_prop]
theorem IsBoundedLinearMap.differentiableOn (h : IsBoundedLinearMap 𝕜 f) : DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : IsBoundedLinearMap 𝕜 f)
Goal: DifferentiableOn 𝕜 f s
-- HasFDerivAtFilter.comp
theorem HasFDerivAtFilter.comp {g : F → G} {g' : F →L[𝕜] G} {L' : Filter F}
    (hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
    HasFDerivAtFilter (g ∘ f) (g'.comp f') x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {L : Filter E} {g : F → G} {g' : F →L[𝕜] G} {L' : Filter F} (hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Filter.Tendsto f L L')
Goal: HasFDerivAtFilter (g ∘ f) (g'.comp f') x L
-- HasFDerivWithinAt.comp
@[fun_prop]
theorem HasFDerivWithinAt.comp {g : F → G} {g' : F →L[𝕜] G} {t : Set F}
    (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
    HasFDerivWithinAt (g ∘ f) (g'.comp f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {s : Set E} {g : F → G} {g' : F →L[𝕜] G} {t : Set F} (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : Set.MapsTo f s t)
Goal: HasFDerivWithinAt (g ∘ f) (g'.comp f') s x
-- HasFDerivAt.comp_hasFDerivWithinAt
@[fun_prop]
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F → G} {g' : F →L[𝕜] G}
    (hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (g ∘ f) (g'.comp f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {s : Set E} {g : F → G} {g' : F →L[𝕜] G} (hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (g ∘ f) (g'.comp f') s x
-- HasFDerivWithinAt.comp_of_mem
@[fun_prop]
theorem HasFDerivWithinAt.comp_of_mem {g : F → G} {g' : F →L[𝕜] G} {t : Set F}
    (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
    (hst : Tendsto f (𝓝[s] x) (𝓝[t] f x)) : HasFDerivWithinAt (g ∘ f) (g'.comp f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {s : Set E} {g : F → G} {g' : F →L[𝕜] G} {t : Set F} (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : Filter.Tendsto f (𝓝[s] x) (𝓝[t] f x))
Goal: HasFDerivWithinAt (g ∘ f) (g'.comp f') s x
-- HasFDerivAt.comp
@[fun_prop]
theorem HasFDerivAt.comp {g : F → G} {g' : F →L[𝕜] G} (hg : HasFDerivAt g g' (f x))
    (hf : HasFDerivAt f f' x) : HasFDerivAt (g ∘ f) (g'.comp f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {g : F → G} {g' : F →L[𝕜] G} (hg : HasFDerivAt g g' (f x)) (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (g ∘ f) (g'.comp f') x
-- DifferentiableWithinAt.comp
@[fun_prop]
theorem DifferentiableWithinAt.comp {g : F → G} {t : Set F}
    (hg : DifferentiableWithinAt 𝕜 g t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x)
    (h : MapsTo f s t) : DifferentiableWithinAt 𝕜 (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G} {t : Set F} (hg : DifferentiableWithinAt 𝕜 g t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x) (h : Set.MapsTo f s t)
Goal: DifferentiableWithinAt 𝕜 (g ∘ f) s x
-- DifferentiableWithinAt.comp'
@[fun_prop]
theorem DifferentiableWithinAt.comp' {g : F → G} {t : Set F}
    (hg : DifferentiableWithinAt 𝕜 g t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x) :
    DifferentiableWithinAt 𝕜 (g ∘ f) (s ∩ f ⁻¹' t) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G} {t : Set F} (hg : DifferentiableWithinAt 𝕜 g t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (g ∘ f) (s ∩ f ⁻¹' t) x
-- DifferentiableAt.comp
@[fun_prop]
theorem DifferentiableAt.comp {g : F → G} (hg : DifferentiableAt 𝕜 g (f x))
    (hf : DifferentiableAt 𝕜 f x) : DifferentiableAt 𝕜 (g ∘ f) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {g : F → G} (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (g ∘ f) x
-- DifferentiableAt.comp_differentiableWithinAt
@[fun_prop]
theorem DifferentiableAt.comp_differentiableWithinAt {g : F → G} (hg : DifferentiableAt 𝕜 g (f x))
    (hf : DifferentiableWithinAt 𝕜 f s x) : DifferentiableWithinAt 𝕜 (g ∘ f) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G} (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (g ∘ f) s x
-- fderivWithin.comp
theorem fderivWithin.comp {g : F → G} {t : Set F} (hg : DifferentiableWithinAt 𝕜 g t (f x))
    (hf : DifferentiableWithinAt 𝕜 f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (g ∘ f) s x = (fderivWithin 𝕜 g t (f x)).comp (fderivWithin 𝕜 f s x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G} {t : Set F} (hg : DifferentiableWithinAt 𝕜 g t (f x)) (hf : DifferentiableWithinAt 𝕜 f s x) (h : Set.MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (g ∘ f) s x = (fderivWithin 𝕜 g t (f x)).comp (fderivWithin 𝕜 f s x)
-- fderivWithin_fderivWithin
theorem fderivWithin_fderivWithin {g : F → G} {f : E → F} {x : E} {y : F} {s : Set E} {t : Set F}
    (hg : DifferentiableWithinAt 𝕜 g t y) (hf : DifferentiableWithinAt 𝕜 f s x) (h : MapsTo f s t)
    (hxs : UniqueDiffWithinAt 𝕜 s x) (hy : f x = y) (v : E) :
    fderivWithin 𝕜 g t y (fderivWithin 𝕜 f s x v) = fderivWithin 𝕜 (g ∘ f) s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {x : E} {y : F} {s : Set E} {t : Set F} (hg : DifferentiableWithinAt 𝕜 g t y) (hf : DifferentiableWithinAt 𝕜 f s x) (h : Set.MapsTo f s t) (hxs : UniqueDiffWithinAt 𝕜 s x) (hy : f x = y) (v : E)
Goal: (fderivWithin 𝕜 g t y) ((fderivWithin 𝕜 f s x) v) = (fderivWithin 𝕜 (g ∘ f) s x) v
-- fderivWithin.comp₃
theorem fderivWithin.comp₃ {g' : G → G'} {g : F → G} {t : Set F} {u : Set G} {y : F} {y' : G}
    (hg' : DifferentiableWithinAt 𝕜 g' u y') (hg : DifferentiableWithinAt 𝕜 g t y)
    (hf : DifferentiableWithinAt 𝕜 f s x) (h2g : MapsTo g t u) (h2f : MapsTo f s t) (h3g : g y = y')
    (h3f : f x = y) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (g' ∘ g ∘ f) s x =
      (fderivWithin 𝕜 g' u y').comp ((fderivWithin 𝕜 g t y).comp (fderivWithin 𝕜 f s x)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] {f : E → F} (x : E) {s : Set E} {g' : G → G'} {g : F → G} {t : Set F} {u : Set G} {y : F} {y' : G} (hg' : DifferentiableWithinAt 𝕜 g' u y') (hg : DifferentiableWithinAt 𝕜 g t y) (hf : DifferentiableWithinAt 𝕜 f s x) (h2g : Set.MapsTo g t u) (h2f : Set.MapsTo f s t) (h3g : g y = y') (h3f : f x = y) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (g' ∘ g ∘ f) s x = (fderivWithin 𝕜 g' u y').comp ((fderivWithin 𝕜 g t y).comp (fderivWithin 𝕜 f s x))
-- fderiv.comp
theorem fderiv.comp {g : F → G} (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableAt 𝕜 f x) :
    fderiv 𝕜 (g ∘ f) x = (fderiv 𝕜 g (f x)).comp (fderiv 𝕜 f x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {g : F → G} (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableAt 𝕜 f x)
Goal: fderiv 𝕜 (g ∘ f) x = (fderiv 𝕜 g (f x)).comp (fderiv 𝕜 f x)
-- fderiv.comp_fderivWithin
theorem fderiv.comp_fderivWithin {g : F → G} (hg : DifferentiableAt 𝕜 g (f x))
    (hf : DifferentiableWithinAt 𝕜 f s x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (g ∘ f) s x = (fderiv 𝕜 g (f x)).comp (fderivWithin 𝕜 f s x) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} (x : E) {s : Set E} {g : F → G} (hg : DifferentiableAt 𝕜 g (f x)) (hf : DifferentiableWithinAt 𝕜 f s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (g ∘ f) s x = (fderiv 𝕜 g (f x)).comp (fderivWithin 𝕜 f s x)
-- DifferentiableOn.comp
@[fun_prop]
theorem DifferentiableOn.comp {g : F → G} {t : Set F} (hg : DifferentiableOn 𝕜 g t)
    (hf : DifferentiableOn 𝕜 f s) (st : MapsTo f s t) : DifferentiableOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {s : Set E} {g : F → G} {t : Set F} (hg : DifferentiableOn 𝕜 g t) (hf : DifferentiableOn 𝕜 f s) (st : Set.MapsTo f s t)
Goal: DifferentiableOn 𝕜 (g ∘ f) s
-- Differentiable.comp
@[fun_prop]
theorem Differentiable.comp {g : F → G} (hg : Differentiable 𝕜 g) (hf : Differentiable 𝕜 f) :
    Differentiable 𝕜 (g ∘ f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {g : F → G} (hg : Differentiable 𝕜 g) (hf : Differentiable 𝕜 f)
Goal: Differentiable 𝕜 (g ∘ f)
-- Differentiable.comp_differentiableOn
@[fun_prop]
theorem Differentiable.comp_differentiableOn {g : F → G} (hg : Differentiable 𝕜 g)
    (hf : DifferentiableOn 𝕜 f s) : DifferentiableOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {s : Set E} {g : F → G} (hg : Differentiable 𝕜 g) (hf : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (g ∘ f) s
-- HasStrictFDerivAt.comp
@[fun_prop]
protected theorem HasStrictFDerivAt.comp {g : F → G} {g' : F →L[𝕜] G}
    (hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => g (f x)) (g'.comp f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {f' : E →L[𝕜] F} (x : E) {g : F → G} {g' : F →L[𝕜] G} (hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => g (f x)) (g'.comp f') x
-- Differentiable.iterate
@[fun_prop]
protected theorem Differentiable.iterate {f : E → E} (hf : Differentiable 𝕜 f) (n : ℕ) :
    Differentiable 𝕜 f^[n] 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → E} (hf : Differentiable 𝕜 f) (n : ℕ)
Goal: Differentiable 𝕜 f^[n]
-- DifferentiableOn.iterate
@[fun_prop]
protected theorem DifferentiableOn.iterate {f : E → E} (hf : DifferentiableOn 𝕜 f s)
    (hs : MapsTo f s s) (n : ℕ) : DifferentiableOn 𝕜 f^[n] s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {f : E → E} (hf : DifferentiableOn 𝕜 f s) (hs : Set.MapsTo f s s) (n : ℕ)
Goal: DifferentiableOn 𝕜 f^[n] s
-- HasFDerivAtFilter.iterate
protected theorem HasFDerivAtFilter.iterate {f : E → E} {f' : E →L[𝕜] E}
    (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : ℕ) :
    HasFDerivAtFilter f^[n] (f' ^ n) x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {L : Filter E} {f : E → E} {f' : E →L[𝕜] E} (hf : HasFDerivAtFilter f f' x L) (hL : Filter.Tendsto f L L) (hx : f x = x) (n : ℕ)
Goal: HasFDerivAtFilter f^[n] (f' ^ n) x L
-- HasFDerivAt.iterate
@[fun_prop]
protected theorem HasFDerivAt.iterate {f : E → E} {f' : E →L[𝕜] E} (hf : HasFDerivAt f f' x)
    (hx : f x = x) (n : ℕ) : HasFDerivAt f^[n] (f' ^ n) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {f : E → E} {f' : E →L[𝕜] E} (hf : HasFDerivAt f f' x) (hx : f x = x) (n : ℕ)
Goal: HasFDerivAt f^[n] (f' ^ n) x
-- HasFDerivWithinAt.iterate
@[fun_prop]
protected theorem HasFDerivWithinAt.iterate {f : E → E} {f' : E →L[𝕜] E}
    (hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : ℕ) :
    HasFDerivWithinAt f^[n] (f' ^ n) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {f : E → E} {f' : E →L[𝕜] E} (hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : Set.MapsTo f s s) (n : ℕ)
Goal: HasFDerivWithinAt f^[n] (f' ^ n) s x
-- HasStrictFDerivAt.iterate
@[fun_prop]
protected theorem HasStrictFDerivAt.iterate {f : E → E} {f' : E →L[𝕜] E}
    (hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : ℕ) :
    HasStrictFDerivAt f^[n] (f' ^ n) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {f : E → E} {f' : E →L[𝕜] E} (hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : ℕ)
Goal: HasStrictFDerivAt f^[n] (f' ^ n) x
-- DifferentiableAt.iterate
@[fun_prop]
protected theorem DifferentiableAt.iterate {f : E → E} (hf : DifferentiableAt 𝕜 f x) (hx : f x = x)
    (n : ℕ) : DifferentiableAt 𝕜 f^[n] x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {f : E → E} (hf : DifferentiableAt 𝕜 f x) (hx : f x = x) (n : ℕ)
Goal: DifferentiableAt 𝕜 f^[n] x
-- DifferentiableWithinAt.iterate
@[fun_prop]
protected theorem DifferentiableWithinAt.iterate {f : E → E} (hf : DifferentiableWithinAt 𝕜 f s x)
    (hx : f x = x) (hs : MapsTo f s s) (n : ℕ) : DifferentiableWithinAt 𝕜 f^[n] s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E} {f : E → E} (hf : DifferentiableWithinAt 𝕜 f s x) (hx : f x = x) (hs : Set.MapsTo f s s) (n : ℕ)
Goal: DifferentiableWithinAt 𝕜 f^[n] s x
-- summable_of_summable_hasFDerivAt_of_isPreconnected
theorem summable_of_summable_hasFDerivAt_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
    (h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable (f · x₀))
    (hx : x ∈ s) : Summable fun n => f n x 
Arguments: {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [NormedSpace 𝕜 F] {f : α → E → F} {f' : α → E → E →L[𝕜] F} {s : Set E} {x₀ x : E} (hu : Summable u) (hs : IsOpen s) (h's : IsPreconnected s) (hf : ∀ (n : α), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ (n : α), ∀ x ∈ s, ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun x => f x x₀) (hx : x ∈ s)
Goal: Summable fun n => f n x
-- summable_of_summable_hasDerivAt_of_isPreconnected
theorem summable_of_summable_hasDerivAt_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
    (h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable (g · y₀))
    (hy : y ∈ t) : Summable fun n => g n y 
Arguments: {α : Type u_1} {𝕜 : Type u_3} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [NormedSpace 𝕜 F] {g g' : α → 𝕜 → F} {t : Set 𝕜} {y₀ y : 𝕜} (hu : Summable u) (ht : IsOpen t) (h't : IsPreconnected t) (hg : ∀ (n : α), ∀ y ∈ t, HasDerivAt (g n) (g' n y) y) (hg' : ∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun x => g x y₀) (hy : y ∈ t)
Goal: Summable fun n => g n y
-- hasFDerivAt_tsum_of_isPreconnected
theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
    (h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀)
    (hx : x ∈ s) : HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x 
Arguments: {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [NormedSpace 𝕜 F] {f : α → E → F} {f' : α → E → E →L[𝕜] F} {s : Set E} {x₀ x : E} (hu : Summable u) (hs : IsOpen s) (h's : IsPreconnected s) (hf : ∀ (n : α), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ (n : α), ∀ x ∈ s, ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀) (hx : x ∈ s)
Goal: HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x
-- hasDerivAt_tsum_of_isPreconnected
theorem hasDerivAt_tsum_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
    (h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun n => g n y₀)
    (hy : y ∈ t) : HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y 
Arguments: {α : Type u_1} {𝕜 : Type u_3} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [NormedSpace 𝕜 F] {g g' : α → 𝕜 → F} {t : Set 𝕜} {y₀ y : 𝕜} (hu : Summable u) (ht : IsOpen t) (h't : IsPreconnected t) (hg : ∀ (n : α), ∀ y ∈ t, HasDerivAt (g n) (g' n y) y) (hg' : ∀ (n : α), ∀ y ∈ t, ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun n => g n y₀) (hy : y ∈ t)
Goal: HasDerivAt (fun z => ∑' (n : α), g n z) (∑' (n : α), g' n y) y
-- summable_of_summable_hasFDerivAt
theorem summable_of_summable_hasFDerivAt (hu : Summable u)
    (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
    (hf0 : Summable fun n => f n x₀) (x : E) : Summable fun n => f n x 
Arguments: {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ} [NormedSpace 𝕜 F] {f : α → E → F} {f' : α → E → E →L[𝕜] F} {x₀ : E} (hu : Summable u) (hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x) (hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E)
Goal: Summable fun n => f n x