ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasFDerivAt.add
@[fun_prop]
nonrec theorem HasFDerivAt.add (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) :
    HasFDerivAt (fun x => f x + g x) (f' + g') 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 g : E → F} {f' g' : E →L[𝕜] F} {x : E} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x)
Goal: HasFDerivAt (fun x => f x + g x) (f' + g') x
-- DifferentiableWithinAt.add
@[fun_prop]
theorem DifferentiableWithinAt.add (hf : DifferentiableWithinAt 𝕜 f s x)
    (hg : DifferentiableWithinAt 𝕜 g s x) : DifferentiableWithinAt 𝕜 (fun y => f y + g y) 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 g : E → F} {x : E} {s : Set E} (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => f y + g y) s x
-- DifferentiableAt.add
@[simp, fun_prop]
theorem DifferentiableAt.add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    DifferentiableAt 𝕜 (fun y => f y + g y) 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 g : E → F} {x : E} (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x)
Goal: DifferentiableAt 𝕜 (fun y => f y + g y) x
-- DifferentiableOn.add
@[fun_prop]
theorem DifferentiableOn.add (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) :
    DifferentiableOn 𝕜 (fun y => f y + g y) 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 g : E → F} {s : Set E} (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s)
Goal: DifferentiableOn 𝕜 (fun y => f y + g y) s
-- Differentiable.add
@[simp, fun_prop]
theorem Differentiable.add (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) :
    Differentiable 𝕜 fun y => f y + g y 
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 g : E → F} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
Goal: Differentiable 𝕜 fun y => f y + g y
-- fderivWithin_add
theorem fderivWithin_add (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x)
    (hg : DifferentiableWithinAt 𝕜 g s x) :
    fderivWithin 𝕜 (fun y => f y + g y) s x = fderivWithin 𝕜 f s x + fderivWithin 𝕜 g 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 g : E → F} {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
Goal: fderivWithin 𝕜 (fun y => f y + g y) s x = fderivWithin 𝕜 f s x + fderivWithin 𝕜 g s x
-- fderiv_add
theorem fderiv_add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    fderiv 𝕜 (fun y => f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g 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 g : E → F} {x : E} (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x)
Goal: fderiv 𝕜 (fun y => f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g x
-- HasStrictFDerivAt.add_const
@[fun_prop]
theorem HasStrictFDerivAt.add_const (hf : HasStrictFDerivAt f f' x) (c : F) :
    HasStrictFDerivAt (fun y => f y + c) 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} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x) (c : F)
Goal: HasStrictFDerivAt (fun y => f y + c) f' x
-- HasFDerivAtFilter.add_const
theorem HasFDerivAtFilter.add_const (hf : HasFDerivAtFilter f f' x L) (c : F) :
    HasFDerivAtFilter (fun y => f y + c) 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] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (hf : HasFDerivAtFilter f f' x L) (c : F)
Goal: HasFDerivAtFilter (fun y => f y + c) f' x L
-- HasFDerivWithinAt.add_const
@[fun_prop]
nonrec theorem HasFDerivWithinAt.add_const (hf : HasFDerivWithinAt f f' s x) (c : F) :
    HasFDerivWithinAt (fun y => f y + c) 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} {f' : E →L[𝕜] F} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (c : F)
Goal: HasFDerivWithinAt (fun y => f y + c) f' s x
-- HasFDerivAt.add_const
@[fun_prop]
nonrec theorem HasFDerivAt.add_const (hf : HasFDerivAt f f' x) (c : F) :
    HasFDerivAt (fun x => f x + c) 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} {f' : E →L[𝕜] F} {x : E} (hf : HasFDerivAt f f' x) (c : F)
Goal: HasFDerivAt (fun x => f x + c) f' x
-- DifferentiableWithinAt.add_const
@[fun_prop]
theorem DifferentiableWithinAt.add_const (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) :
    DifferentiableWithinAt 𝕜 (fun y => f y + c) 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} (hf : DifferentiableWithinAt 𝕜 f s x) (c : F)
Goal: DifferentiableWithinAt 𝕜 (fun y => f y + c) s x
-- differentiableWithinAt_add_const_iff
@[simp]
theorem differentiableWithinAt_add_const_iff (c : F) :
    DifferentiableWithinAt 𝕜 (fun y => f y + c) s x ↔ 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} (c : F)
Goal: DifferentiableWithinAt 𝕜 (fun y => f y + c) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableAt.add_const
@[fun_prop]
theorem DifferentiableAt.add_const (hf : DifferentiableAt 𝕜 f x) (c : F) :
    DifferentiableAt 𝕜 (fun y => f y + c) 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} (hf : DifferentiableAt 𝕜 f x) (c : F)
Goal: DifferentiableAt 𝕜 (fun y => f y + c) x
-- differentiableAt_add_const_iff
@[simp]
theorem differentiableAt_add_const_iff (c : F) :
    DifferentiableAt 𝕜 (fun y => f y + c) x ↔ 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} (c : F)
Goal: DifferentiableAt 𝕜 (fun y => f y + c) x ↔ DifferentiableAt 𝕜 f x
-- DifferentiableOn.add_const
@[fun_prop]
theorem DifferentiableOn.add_const (hf : DifferentiableOn 𝕜 f s) (c : F) :
    DifferentiableOn 𝕜 (fun y => f y + c) 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} (hf : DifferentiableOn 𝕜 f s) (c : F)
Goal: DifferentiableOn 𝕜 (fun y => f y + c) s
-- differentiableOn_add_const_iff
@[simp]
theorem differentiableOn_add_const_iff (c : F) :
    DifferentiableOn 𝕜 (fun y => f y + c) s ↔ 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} (c : F)
Goal: DifferentiableOn 𝕜 (fun y => f y + c) s ↔ DifferentiableOn 𝕜 f s
-- Differentiable.add_const
@[fun_prop]
theorem Differentiable.add_const (hf : Differentiable 𝕜 f) (c : F) :
    Differentiable 𝕜 fun y => f y + c 
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} (hf : Differentiable 𝕜 f) (c : F)
Goal: Differentiable 𝕜 fun y => f y + c
-- differentiable_add_const_iff
@[simp]
theorem differentiable_add_const_iff (c : F) :
    (Differentiable 𝕜 fun y => f y + c) ↔ 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} (c : F)
Goal: (Differentiable 𝕜 fun y => f y + c) ↔ Differentiable 𝕜 f
-- fderivWithin_add_const
theorem fderivWithin_add_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    fderivWithin 𝕜 (fun y => f y + c) s x = 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] {f : E → F} {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: fderivWithin 𝕜 (fun y => f y + c) s x = fderivWithin 𝕜 f s x
-- fderiv_add_const
theorem fderiv_add_const (c : F) : fderiv 𝕜 (fun y => f y + c) x = 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] {f : E → F} {x : E} (c : F)
Goal: fderiv 𝕜 (fun y => f y + c) x = fderiv 𝕜 f x
-- HasStrictFDerivAt.const_add
@[fun_prop]
theorem HasStrictFDerivAt.const_add (hf : HasStrictFDerivAt f f' x) (c : F) :
    HasStrictFDerivAt (fun y => c + f y) 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} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x) (c : F)
Goal: HasStrictFDerivAt (fun y => c + f y) f' x
-- HasFDerivAtFilter.const_add
theorem HasFDerivAtFilter.const_add (hf : HasFDerivAtFilter f f' x L) (c : F) :
    HasFDerivAtFilter (fun y => c + f y) 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] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (hf : HasFDerivAtFilter f f' x L) (c : F)
Goal: HasFDerivAtFilter (fun y => c + f y) f' x L
-- HasFDerivWithinAt.const_add
@[fun_prop]
nonrec theorem HasFDerivWithinAt.const_add (hf : HasFDerivWithinAt f f' s x) (c : F) :
    HasFDerivWithinAt (fun y => c + f y) 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} {f' : E →L[𝕜] F} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (c : F)
Goal: HasFDerivWithinAt (fun y => c + f y) f' s x
-- HasFDerivAt.const_add
@[fun_prop]
nonrec theorem HasFDerivAt.const_add (hf : HasFDerivAt f f' x) (c : F) :
    HasFDerivAt (fun x => c + f x) 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} {f' : E →L[𝕜] F} {x : E} (hf : HasFDerivAt f f' x) (c : F)
Goal: HasFDerivAt (fun x => c + f x) f' x
-- DifferentiableWithinAt.const_add
@[fun_prop]
theorem DifferentiableWithinAt.const_add (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) :
    DifferentiableWithinAt 𝕜 (fun y => c + f y) 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} (hf : DifferentiableWithinAt 𝕜 f s x) (c : F)
Goal: DifferentiableWithinAt 𝕜 (fun y => c + f y) s x
-- differentiableWithinAt_const_add_iff
@[simp]
theorem differentiableWithinAt_const_add_iff (c : F) :
    DifferentiableWithinAt 𝕜 (fun y => c + f y) s x ↔ 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} (c : F)
Goal: DifferentiableWithinAt 𝕜 (fun y => c + f y) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableAt.const_add
@[fun_prop]
theorem DifferentiableAt.const_add (hf : DifferentiableAt 𝕜 f x) (c : F) :
    DifferentiableAt 𝕜 (fun y => c + f y) 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} (hf : DifferentiableAt 𝕜 f x) (c : F)
Goal: DifferentiableAt 𝕜 (fun y => c + f y) x
-- differentiableAt_const_add_iff
@[simp]
theorem differentiableAt_const_add_iff (c : F) :
    DifferentiableAt 𝕜 (fun y => c + f y) x ↔ 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} (c : F)
Goal: DifferentiableAt 𝕜 (fun y => c + f y) x ↔ DifferentiableAt 𝕜 f x
-- DifferentiableOn.const_add
@[fun_prop]
theorem DifferentiableOn.const_add (hf : DifferentiableOn 𝕜 f s) (c : F) :
    DifferentiableOn 𝕜 (fun y => c + f y) 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} (hf : DifferentiableOn 𝕜 f s) (c : F)
Goal: DifferentiableOn 𝕜 (fun y => c + f y) s
-- differentiableOn_const_add_iff
@[simp]
theorem differentiableOn_const_add_iff (c : F) :
    DifferentiableOn 𝕜 (fun y => c + f y) s ↔ 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} (c : F)
Goal: DifferentiableOn 𝕜 (fun y => c + f y) s ↔ DifferentiableOn 𝕜 f s
-- Differentiable.const_add
@[fun_prop]
theorem Differentiable.const_add (hf : Differentiable 𝕜 f) (c : F) :
    Differentiable 𝕜 fun y => c + f y 
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} (hf : Differentiable 𝕜 f) (c : F)
Goal: Differentiable 𝕜 fun y => c + f y
-- differentiable_const_add_iff
@[simp]
theorem differentiable_const_add_iff (c : F) :
    (Differentiable 𝕜 fun y => c + f y) ↔ 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} (c : F)
Goal: (Differentiable 𝕜 fun y => c + f y) ↔ Differentiable 𝕜 f
-- fderivWithin_const_add
theorem fderivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    fderivWithin 𝕜 (fun y => c + f y) s x = 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] {f : E → F} {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: fderivWithin 𝕜 (fun y => c + f y) s x = fderivWithin 𝕜 f s x
-- fderiv_const_add
theorem fderiv_const_add (c : F) : fderiv 𝕜 (fun y => c + f y) x = 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] {f : E → F} {x : E} (c : F)
Goal: fderiv 𝕜 (fun y => c + f y) x = fderiv 𝕜 f x
-- HasStrictFDerivAt.sum
@[fun_prop]
theorem HasStrictFDerivAt.sum (h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x) :
    HasStrictFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) 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] {x : E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} (h : ∀ i ∈ u, HasStrictFDerivAt (A i) (A' i) x)
Goal: HasStrictFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x
-- HasFDerivAtFilter.sum
theorem HasFDerivAtFilter.sum (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L) :
    HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) 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] {x : E} {L : Filter E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} (h : ∀ i ∈ u, HasFDerivAtFilter (A i) (A' i) x L)
Goal: HasFDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L
-- HasFDerivWithinAt.sum
@[fun_prop]
theorem HasFDerivWithinAt.sum (h : ∀ i ∈ u, HasFDerivWithinAt (A i) (A' i) s x) :
    HasFDerivWithinAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) 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] {x : E} {s : Set E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} (h : ∀ i ∈ u, HasFDerivWithinAt (A i) (A' i) s x)
Goal: HasFDerivWithinAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) s x
-- HasFDerivAt.sum
@[fun_prop]
theorem HasFDerivAt.sum (h : ∀ i ∈ u, HasFDerivAt (A i) (A' i) x) :
    HasFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) 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] {x : E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} {A' : ι → E →L[𝕜] F} (h : ∀ i ∈ u, HasFDerivAt (A i) (A' i) x)
Goal: HasFDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x
-- DifferentiableWithinAt.sum
@[fun_prop]
theorem DifferentiableWithinAt.sum (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) :
    DifferentiableWithinAt 𝕜 (fun y => ∑ i ∈ u, A i y) 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] {x : E} {s : Set E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => ∑ i ∈ u, A i y) s x
-- DifferentiableAt.sum
@[simp, fun_prop]
theorem DifferentiableAt.sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) :
    DifferentiableAt 𝕜 (fun y => ∑ i ∈ u, A i y) 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] {x : E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x)
Goal: DifferentiableAt 𝕜 (fun y => ∑ i ∈ u, A i y) x
-- DifferentiableOn.sum
@[fun_prop]
theorem DifferentiableOn.sum (h : ∀ i ∈ u, DifferentiableOn 𝕜 (A i) s) :
    DifferentiableOn 𝕜 (fun y => ∑ i ∈ u, A i y) 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] {s : Set E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (h : ∀ i ∈ u, DifferentiableOn 𝕜 (A i) s)
Goal: DifferentiableOn 𝕜 (fun y => ∑ i ∈ u, A i y) s
-- Differentiable.sum
@[simp, fun_prop]
theorem Differentiable.sum (h : ∀ i ∈ u, Differentiable 𝕜 (A i)) :
    Differentiable 𝕜 fun y => ∑ i ∈ u, A i y 
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] {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (h : ∀ i ∈ u, Differentiable 𝕜 (A i))
Goal: Differentiable 𝕜 fun y => ∑ i ∈ u, A i y
-- fderivWithin_sum
theorem fderivWithin_sum (hxs : UniqueDiffWithinAt 𝕜 s x)
    (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) :
    fderivWithin 𝕜 (fun y => ∑ i ∈ u, A i y) s x = ∑ i ∈ u, fderivWithin 𝕜 (A i) 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] {x : E} {s : Set E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x)
Goal: fderivWithin 𝕜 (fun y => ∑ i ∈ u, A i y) s x = ∑ i ∈ u, fderivWithin 𝕜 (A i) s x
-- fderiv_sum
theorem fderiv_sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) :
    fderiv 𝕜 (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, fderiv 𝕜 (A i) 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] {x : E} {ι : Type u_6} {u : Finset ι} {A : ι → E → F} (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x)
Goal: fderiv 𝕜 (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, fderiv 𝕜 (A i) x
-- HasStrictFDerivAt.neg
@[fun_prop]
theorem HasStrictFDerivAt.neg (h : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => -f x) (-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} {f' : E →L[𝕜] F} {x : E} (h : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => -f x) (-f') x
-- HasFDerivAtFilter.neg
theorem HasFDerivAtFilter.neg (h : HasFDerivAtFilter f f' x L) :
    HasFDerivAtFilter (fun x => -f x) (-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] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (h : HasFDerivAtFilter f f' x L)
Goal: HasFDerivAtFilter (fun x => -f x) (-f') x L
-- HasFDerivWithinAt.neg
@[fun_prop]
nonrec theorem HasFDerivWithinAt.neg (h : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => -f x) (-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} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => -f x) (-f') s x
-- HasFDerivAt.neg
@[fun_prop]
nonrec theorem HasFDerivAt.neg (h : HasFDerivAt f f' x) : HasFDerivAt (fun x => -f x) (-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} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => -f x) (-f') x
-- DifferentiableWithinAt.neg
@[fun_prop]
theorem DifferentiableWithinAt.neg (h : DifferentiableWithinAt 𝕜 f s x) :
    DifferentiableWithinAt 𝕜 (fun y => -f y) 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 : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => -f y) s x
-- differentiableWithinAt_neg_iff
@[simp]
theorem differentiableWithinAt_neg_iff :
    DifferentiableWithinAt 𝕜 (fun y => -f y) s x ↔ 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}
Goal: DifferentiableWithinAt 𝕜 (fun y => -f y) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableAt.neg
@[fun_prop]
theorem DifferentiableAt.neg (h : DifferentiableAt 𝕜 f x) : DifferentiableAt 𝕜 (fun y => -f y) 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 : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun y => -f y) x
-- differentiableAt_neg_iff
@[simp]
theorem differentiableAt_neg_iff : DifferentiableAt 𝕜 (fun y => -f y) x ↔ 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}
Goal: DifferentiableAt 𝕜 (fun y => -f y) x ↔ DifferentiableAt 𝕜 f x