ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- DifferentiableOn.neg
@[fun_prop]
theorem DifferentiableOn.neg (h : DifferentiableOn 𝕜 f s) : DifferentiableOn 𝕜 (fun y => -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} (h : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun y => -f y) s
-- differentiableOn_neg_iff
@[simp]
theorem differentiableOn_neg_iff : DifferentiableOn 𝕜 (fun y => -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}
Goal: DifferentiableOn 𝕜 (fun y => -f y) s ↔ DifferentiableOn 𝕜 f s
-- Differentiable.neg
@[fun_prop]
theorem Differentiable.neg (h : Differentiable 𝕜 f) : Differentiable 𝕜 fun y => -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} (h : Differentiable 𝕜 f)
Goal: Differentiable 𝕜 fun y => -f y
-- differentiable_neg_iff
@[simp]
theorem differentiable_neg_iff : (Differentiable 𝕜 fun y => -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}
Goal: (Differentiable 𝕜 fun y => -f y) ↔ Differentiable 𝕜 f
-- fderivWithin_neg
theorem fderivWithin_neg (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun y => -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)
Goal: fderivWithin 𝕜 (fun y => -f y) s x = -fderivWithin 𝕜 f s x
-- fderiv_neg
@[simp]
theorem fderiv_neg : fderiv 𝕜 (fun y => -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}
Goal: fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x
-- HasStrictFDerivAt.sub
@[fun_prop]
theorem HasStrictFDerivAt.sub (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
    HasStrictFDerivAt (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 : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
Goal: HasStrictFDerivAt (fun x => f x - g x) (f' - g') x
-- HasFDerivAtFilter.sub
theorem HasFDerivAtFilter.sub (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L) :
    HasFDerivAtFilter (fun x => f x - g x) (f' - g') 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 g : E → F} {f' g' : E →L[𝕜] F} {x : E} {L : Filter E} (hf : HasFDerivAtFilter f f' x L) (hg : HasFDerivAtFilter g g' x L)
Goal: HasFDerivAtFilter (fun x => f x - g x) (f' - g') x L
-- HasFDerivWithinAt.sub
@[fun_prop]
nonrec theorem HasFDerivWithinAt.sub (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt g g' s x) : HasFDerivWithinAt (fun x => f x - g x) (f' - 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} {f' g' : E →L[𝕜] F} {x : E} {s : Set E} (hf : HasFDerivWithinAt f f' s x) (hg : HasFDerivWithinAt g g' s x)
Goal: HasFDerivWithinAt (fun x => f x - g x) (f' - g') s x
-- HasFDerivAt.sub
@[fun_prop]
nonrec theorem HasFDerivAt.sub (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.sub
@[fun_prop]
theorem DifferentiableWithinAt.sub (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.sub
@[simp, fun_prop]
theorem DifferentiableAt.sub (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
-- DifferentiableAt.add_iff_left
@[simp]
lemma DifferentiableAt.add_iff_left (hg : DifferentiableAt 𝕜 g x) :
    DifferentiableAt 𝕜 (fun y => f y + g 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 g : E → F} {x : E} (hg : DifferentiableAt 𝕜 g x)
Goal: DifferentiableAt 𝕜 (fun y => f y + g y) x ↔ DifferentiableAt 𝕜 f x
-- DifferentiableAt.add_iff_right
@[simp]
lemma DifferentiableAt.add_iff_right (hg : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun y => f y + g y) x ↔ DifferentiableAt 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} (hg : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun y => f y + g y) x ↔ DifferentiableAt 𝕜 g x
-- DifferentiableAt.sub_iff_left
@[simp]
lemma DifferentiableAt.sub_iff_left (hg : DifferentiableAt 𝕜 g x) :
    DifferentiableAt 𝕜 (fun y => f y - g 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 g : E → F} {x : E} (hg : DifferentiableAt 𝕜 g x)
Goal: DifferentiableAt 𝕜 (fun y => f y - g y) x ↔ DifferentiableAt 𝕜 f x
-- DifferentiableAt.sub_iff_right
@[simp]
lemma DifferentiableAt.sub_iff_right (hg : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun y => f y - g y) x ↔ DifferentiableAt 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} (hg : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun y => f y - g y) x ↔ DifferentiableAt 𝕜 g x
-- DifferentiableOn.sub
@[fun_prop]
theorem DifferentiableOn.sub (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
-- DifferentiableOn.add_iff_left
@[simp]
lemma DifferentiableOn.add_iff_left (hg : DifferentiableOn 𝕜 g s) :
    DifferentiableOn 𝕜 (fun y => f y + g 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 g : E → F} {s : Set E} (hg : DifferentiableOn 𝕜 g s)
Goal: DifferentiableOn 𝕜 (fun y => f y + g y) s ↔ DifferentiableOn 𝕜 f s
-- DifferentiableOn.add_iff_right
@[simp]
lemma DifferentiableOn.add_iff_right (hg : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun y => f y + g y) s ↔ DifferentiableOn 𝕜 g 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} (hg : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun y => f y + g y) s ↔ DifferentiableOn 𝕜 g s
-- DifferentiableOn.sub_iff_left
@[simp]
lemma DifferentiableOn.sub_iff_left (hg : DifferentiableOn 𝕜 g s) :
    DifferentiableOn 𝕜 (fun y => f y - g 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 g : E → F} {s : Set E} (hg : DifferentiableOn 𝕜 g s)
Goal: DifferentiableOn 𝕜 (fun y => f y - g y) s ↔ DifferentiableOn 𝕜 f s
-- DifferentiableOn.sub_iff_right
@[simp]
lemma DifferentiableOn.sub_iff_right (hg : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun y => f y - g y) s ↔ DifferentiableOn 𝕜 g 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} (hg : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun y => f y - g y) s ↔ DifferentiableOn 𝕜 g s
-- Differentiable.sub
@[simp, fun_prop]
theorem Differentiable.sub (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
-- Differentiable.add_iff_left
@[simp]
lemma Differentiable.add_iff_left (hg : Differentiable 𝕜 g) :
    Differentiable 𝕜 (fun y => f y + g 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 g : E → F} (hg : Differentiable 𝕜 g)
Goal: (Differentiable 𝕜 fun y => f y + g y) ↔ Differentiable 𝕜 f
-- Differentiable.add_iff_right
@[simp]
lemma Differentiable.add_iff_right (hg : Differentiable 𝕜 f) :
    Differentiable 𝕜 (fun y => f y + g y) ↔ Differentiable 𝕜 g 
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} (hg : Differentiable 𝕜 f)
Goal: (Differentiable 𝕜 fun y => f y + g y) ↔ Differentiable 𝕜 g
-- Differentiable.sub_iff_left
@[simp]
lemma Differentiable.sub_iff_left (hg : Differentiable 𝕜 g) :
    Differentiable 𝕜 (fun y => f y - g 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 g : E → F} (hg : Differentiable 𝕜 g)
Goal: (Differentiable 𝕜 fun y => f y - g y) ↔ Differentiable 𝕜 f
-- Differentiable.sub_iff_right
@[simp]
lemma Differentiable.sub_iff_right (hg : Differentiable 𝕜 f) :
    Differentiable 𝕜 (fun y => f y - g y) ↔ Differentiable 𝕜 g 
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} (hg : Differentiable 𝕜 f)
Goal: (Differentiable 𝕜 fun y => f y - g y) ↔ Differentiable 𝕜 g
-- fderivWithin_sub
theorem fderivWithin_sub (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_sub
theorem fderiv_sub (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.sub_const
@[fun_prop]
theorem HasStrictFDerivAt.sub_const (hf : HasStrictFDerivAt f f' x) (c : F) :
    HasStrictFDerivAt (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 : HasStrictFDerivAt f f' x) (c : F)
Goal: HasStrictFDerivAt (fun x => f x - c) f' x
-- HasFDerivAtFilter.sub_const
theorem HasFDerivAtFilter.sub_const (hf : HasFDerivAtFilter f f' x L) (c : F) :
    HasFDerivAtFilter (fun x => f x - 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 x => f x - c) f' x L
-- HasFDerivWithinAt.sub_const
@[fun_prop]
nonrec theorem HasFDerivWithinAt.sub_const (hf : HasFDerivWithinAt f f' s x) (c : F) :
    HasFDerivWithinAt (fun x => f x - 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 x => f x - c) f' s x
-- HasFDerivAt.sub_const
@[fun_prop]
nonrec theorem HasFDerivAt.sub_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
-- hasStrictFDerivAt_sub_const
@[fun_prop]
theorem hasStrictFDerivAt_sub_const {x : F} (c : F) : HasStrictFDerivAt (· - c) (id 𝕜 F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : F} (c : F)
Goal: HasStrictFDerivAt (fun x => x - c) (ContinuousLinearMap.id 𝕜 F) x
-- hasFDerivAt_sub_const
@[fun_prop]
theorem hasFDerivAt_sub_const {x : F} (c : F) : HasFDerivAt (· - c) (id 𝕜 F) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : F} (c : F)
Goal: HasFDerivAt (fun x => x - c) (ContinuousLinearMap.id 𝕜 F) x
-- DifferentiableWithinAt.sub_const
@[fun_prop]
theorem DifferentiableWithinAt.sub_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_sub_const_iff
@[simp]
theorem differentiableWithinAt_sub_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.sub_const
@[fun_prop]
theorem DifferentiableAt.sub_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_sub_const_iff
@[deprecated DifferentiableAt.sub_iff_left (since := "2024-07-11")]
theorem differentiableAt_sub_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.sub_const
@[fun_prop]
theorem DifferentiableOn.sub_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_sub_const_iff
@[deprecated DifferentiableOn.sub_iff_left (since := "2024-07-11")]
theorem differentiableOn_sub_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.sub_const
@[fun_prop]
theorem Differentiable.sub_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_sub_const_iff
@[deprecated Differentiable.sub_iff_left (since := "2024-07-11")]
theorem differentiable_sub_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_sub_const
theorem fderivWithin_sub_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_sub_const
theorem fderiv_sub_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_sub
@[fun_prop]
theorem HasStrictFDerivAt.const_sub (hf : HasStrictFDerivAt f f' x) (c : F) :
    HasStrictFDerivAt (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 : HasStrictFDerivAt f f' x) (c : F)
Goal: HasStrictFDerivAt (fun x => c - f x) (-f') x
-- HasFDerivAtFilter.const_sub
theorem HasFDerivAtFilter.const_sub (hf : HasFDerivAtFilter f f' x L) (c : F) :
    HasFDerivAtFilter (fun x => c - 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} (hf : HasFDerivAtFilter f f' x L) (c : F)
Goal: HasFDerivAtFilter (fun x => c - f x) (-f') x L
-- HasFDerivWithinAt.const_sub
@[fun_prop]
nonrec theorem HasFDerivWithinAt.const_sub (hf : HasFDerivWithinAt f f' s x) (c : F) :
    HasFDerivWithinAt (fun x => c - 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} (hf : HasFDerivWithinAt f f' s x) (c : F)
Goal: HasFDerivWithinAt (fun x => c - f x) (-f') s x
-- HasFDerivAt.const_sub
@[fun_prop]
nonrec theorem HasFDerivAt.const_sub (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_sub
@[fun_prop]
theorem DifferentiableWithinAt.const_sub (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_sub_iff
@[simp]
theorem differentiableWithinAt_const_sub_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_sub
@[fun_prop]
theorem DifferentiableAt.const_sub (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_sub_iff
@[deprecated DifferentiableAt.sub_iff_right (since := "2024-07-11")]
theorem differentiableAt_const_sub_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_sub
@[fun_prop]
theorem DifferentiableOn.const_sub (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