ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousLinearMap.hasFDerivAt_of_bilinear
@[fun_prop]
theorem ContinuousLinearMap.hasFDerivAt_of_bilinear {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E}
    {g' : G' →L[𝕜] F} {x : G'} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) :
    HasFDerivAt (fun y => B (f y) (g y)) (B.precompR G' (f x) g' + B.precompL G' f' (g x)) 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'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x)
Goal: HasFDerivAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x
-- ContinuousLinearMap.hasStrictFDerivAt_of_bilinear
@[fun_prop]
theorem ContinuousLinearMap.hasStrictFDerivAt_of_bilinear
    {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E}
    {g' : G' →L[𝕜] F} {x : G'} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
    HasStrictFDerivAt (fun y => B (f y) (g y))
      (B.precompR G' (f x) g' + B.precompL G' f' (g x)) 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'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
Goal: HasStrictFDerivAt (fun y => (B (f y)) (g y))
  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x
-- ContinuousLinearMap.fderivWithin_of_bilinear
theorem ContinuousLinearMap.fderivWithin_of_bilinear {f : G' → E} {g : G' → F} {x : G'} {s : Set G'}
    (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
    (hs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun y => B (f y) (g y)) s x =
      B.precompR G' (f x) (fderivWithin 𝕜 g s x) + B.precompL G' (fderivWithin 𝕜 f s x) (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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {x : G'} {s : Set G'} (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x) (hs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun y => (B (f y)) (g y)) s x =
  ((ContinuousLinearMap.precompR G' B) (f x)) (fderivWithin 𝕜 g s x) +
    ((ContinuousLinearMap.precompL G' B) (fderivWithin 𝕜 f s x)) (g x)
-- ContinuousLinearMap.fderiv_of_bilinear
theorem ContinuousLinearMap.fderiv_of_bilinear {f : G' → E} {g : G' → F} {x : G'}
    (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    fderiv 𝕜 (fun y => B (f y) (g y)) x =
      B.precompR G' (f x) (fderiv 𝕜 g x) + B.precompL G' (fderiv 𝕜 f x) (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] {G : Type u_4} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {G' : Type u_5} [NormedAddCommGroup G'] [NormedSpace 𝕜 G'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {x : G'} (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x)
Goal: fderiv 𝕜 (fun y => (B (f y)) (g y)) x =
  ((ContinuousLinearMap.precompR G' B) (f x)) (fderiv 𝕜 g x) +
    ((ContinuousLinearMap.precompL G' B) (fderiv 𝕜 f x)) (g x)
-- dslope_same
@[simp]
theorem dslope_same (f : 𝕜 → E) (a : 𝕜) : dslope f a a = deriv f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 → E) (a : 𝕜)
Goal: dslope f a a = deriv f a
-- dslope_of_ne
theorem dslope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a b = slope f a b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : 𝕜} (f : 𝕜 → E) (h : b ≠ a)
Goal: dslope f a b = slope f a b
-- ContinuousLinearMap.dslope_comp
theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
    (f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
    dslope (f ∘ g) a b = f (dslope g a b) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a)
Goal: dslope (⇑f ∘ g) a b = f (dslope g a b)
-- eqOn_dslope_slope
theorem eqOn_dslope_slope (f : 𝕜 → E) (a : 𝕜) : EqOn (dslope f a) (slope f a) {a}ᶜ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 → E) (a : 𝕜)
Goal: Set.EqOn (dslope f a) (slope f a) {a}ᶜ
-- dslope_eventuallyEq_slope_of_ne
theorem dslope_eventuallyEq_slope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a =ᶠ[𝓝 b] slope f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : 𝕜} (f : 𝕜 → E) (h : b ≠ a)
Goal: dslope f a =ᶠ[𝓝 b] slope f a
-- dslope_eventuallyEq_slope_punctured_nhds
theorem dslope_eventuallyEq_slope_punctured_nhds (f : 𝕜 → E) : dslope f a =ᶠ[𝓝[≠] a] slope f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a : 𝕜} (f : 𝕜 → E)
Goal: dslope f a =ᶠ[𝓝[≠] a] slope f a
-- sub_smul_dslope
@[simp]
theorem sub_smul_dslope (f : 𝕜 → E) (a b : 𝕜) : (b - a) • dslope f a b = f b - f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 → E) (a b : 𝕜)
Goal: (b - a) • dslope f a b = f b - f a
-- dslope_sub_smul_of_ne
theorem dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) :
    dslope (fun x => (x - a) • f x) a b = f b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {a b : 𝕜} (f : 𝕜 → E) (h : b ≠ a)
Goal: dslope (fun x => (x - a) • f x) a b = f b
-- eqOn_dslope_sub_smul
theorem eqOn_dslope_sub_smul (f : 𝕜 → E) (a : 𝕜) :
    EqOn (dslope (fun x => (x - a) • f x) a) f {a}ᶜ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (f : 𝕜 → E) (a : 𝕜)
Goal: Set.EqOn (dslope (fun x => (x - a) • f x) a) f {a}ᶜ
-- dslope_sub_smul
theorem dslope_sub_smul [DecidableEq 𝕜] (f : 𝕜 → E) (a : 𝕜) :
    dslope (fun x => (x - a) • f x) a = update f a (deriv (fun x => (x - a) • f x) a) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [DecidableEq 𝕜] (f : 𝕜 → E) (a : 𝕜)
Goal: dslope (fun x => (x - a) • f x) a = Function.update f a (deriv (fun x => (x - a) • f x) a)
-- continuousAt_dslope_same
@[simp]
theorem continuousAt_dslope_same : ContinuousAt (dslope f a) a ↔ DifferentiableAt 𝕜 f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a : 𝕜}
Goal: ContinuousAt (dslope f a) a ↔ DifferentiableAt 𝕜 f a
-- ContinuousWithinAt.of_dslope
theorem ContinuousWithinAt.of_dslope (h : ContinuousWithinAt (dslope f a) s b) :
    ContinuousWithinAt f s b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} {s : Set 𝕜} (h : ContinuousWithinAt (dslope f a) s b)
Goal: ContinuousWithinAt f s b
-- ContinuousAt.of_dslope
theorem ContinuousAt.of_dslope (h : ContinuousAt (dslope f a) b) : ContinuousAt f b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} (h : ContinuousAt (dslope f a) b)
Goal: ContinuousAt f b
-- ContinuousOn.of_dslope
theorem ContinuousOn.of_dslope (h : ContinuousOn (dslope f a) s) : ContinuousOn f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a : 𝕜} {s : Set 𝕜} (h : ContinuousOn (dslope f a) s)
Goal: ContinuousOn f s
-- continuousWithinAt_dslope_of_ne
theorem continuousWithinAt_dslope_of_ne (h : b ≠ a) :
    ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} {s : Set 𝕜} (h : b ≠ a)
Goal: ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b
-- continuousAt_dslope_of_ne
theorem continuousAt_dslope_of_ne (h : b ≠ a) : ContinuousAt (dslope f a) b ↔ ContinuousAt f b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} (h : b ≠ a)
Goal: ContinuousAt (dslope f a) b ↔ ContinuousAt f b
-- continuousOn_dslope
theorem continuousOn_dslope (h : s ∈ 𝓝 a) :
    ContinuousOn (dslope f a) s ↔ ContinuousOn f s ∧ DifferentiableAt 𝕜 f a 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a : 𝕜} {s : Set 𝕜} (h : s ∈ 𝓝 a)
Goal: ContinuousOn (dslope f a) s ↔ ContinuousOn f s ∧ DifferentiableAt 𝕜 f a
-- DifferentiableWithinAt.of_dslope
theorem DifferentiableWithinAt.of_dslope (h : DifferentiableWithinAt 𝕜 (dslope f a) s b) :
    DifferentiableWithinAt 𝕜 f s b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} {s : Set 𝕜} (h : DifferentiableWithinAt 𝕜 (dslope f a) s b)
Goal: DifferentiableWithinAt 𝕜 f s b
-- DifferentiableAt.of_dslope
theorem DifferentiableAt.of_dslope (h : DifferentiableAt 𝕜 (dslope f a) b) :
    DifferentiableAt 𝕜 f b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} (h : DifferentiableAt 𝕜 (dslope f a) b)
Goal: DifferentiableAt 𝕜 f b
-- DifferentiableOn.of_dslope
theorem DifferentiableOn.of_dslope (h : DifferentiableOn 𝕜 (dslope f a) s) :
    DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a : 𝕜} {s : Set 𝕜} (h : DifferentiableOn 𝕜 (dslope f a) s)
Goal: DifferentiableOn 𝕜 f s
-- differentiableWithinAt_dslope_of_ne
theorem differentiableWithinAt_dslope_of_ne (h : b ≠ a) :
    DifferentiableWithinAt 𝕜 (dslope f a) s b ↔ DifferentiableWithinAt 𝕜 f s b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} {s : Set 𝕜} (h : b ≠ a)
Goal: DifferentiableWithinAt 𝕜 (dslope f a) s b ↔ DifferentiableWithinAt 𝕜 f s b
-- differentiableOn_dslope_of_nmem
theorem differentiableOn_dslope_of_nmem (h : a ∉ s) :
    DifferentiableOn 𝕜 (dslope f a) s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a : 𝕜} {s : Set 𝕜} (h : a ∉ s)
Goal: DifferentiableOn 𝕜 (dslope f a) s ↔ DifferentiableOn 𝕜 f s
-- differentiableAt_dslope_of_ne
theorem differentiableAt_dslope_of_ne (h : b ≠ a) :
    DifferentiableAt 𝕜 (dslope f a) b ↔ DifferentiableAt 𝕜 f b 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : 𝕜 → E} {a b : 𝕜} (h : b ≠ a)
Goal: DifferentiableAt 𝕜 (dslope f a) b ↔ DifferentiableAt 𝕜 f b
-- DiffContOnCl.differentiableOn
structure DiffContOnCl (f : E → F) (s : Set E) : Prop where
  protected differentiableOn : DifferentiableOn 𝕜 f s
  protected continuousOn : ContinuousOn f (closure s)
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (self : DiffContOnCl 𝕜 f s)
Goal: DifferentiableOn 𝕜 f s
-- DiffContOnCl.continuousOn
structure DiffContOnCl (f : E → F) (s : Set E) : Prop where
  protected differentiableOn : DifferentiableOn 𝕜 f s
  protected continuousOn : ContinuousOn f (closure s)
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (self : DiffContOnCl 𝕜 f s)
Goal: ContinuousOn f (closure s)
-- DifferentiableOn.diffContOnCl
theorem DifferentiableOn.diffContOnCl (h : DifferentiableOn 𝕜 f (closure s)) : DiffContOnCl 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : DifferentiableOn 𝕜 f (closure s))
Goal: DiffContOnCl 𝕜 f s
-- Differentiable.diffContOnCl
theorem Differentiable.diffContOnCl (h : Differentiable 𝕜 f) : DiffContOnCl 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (h : Differentiable 𝕜 f)
Goal: DiffContOnCl 𝕜 f s
-- IsClosed.diffContOnCl_iff
theorem IsClosed.diffContOnCl_iff (hs : IsClosed s) : DiffContOnCl 𝕜 f s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hs : IsClosed s)
Goal: DiffContOnCl 𝕜 f s ↔ DifferentiableOn 𝕜 f s
-- diffContOnCl_univ
theorem diffContOnCl_univ : DiffContOnCl 𝕜 f univ ↔ Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F}
Goal: DiffContOnCl 𝕜 f Set.univ ↔ Differentiable 𝕜 f
-- diffContOnCl_const
theorem diffContOnCl_const {c : F} : DiffContOnCl 𝕜 (fun _ : E => c) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {s : Set E} {c : F}
Goal: DiffContOnCl 𝕜 (fun x => c) s
-- DiffContOnCl.comp
theorem comp {g : G → E} {t : Set G} (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g t)
    (h : MapsTo g t s) : DiffContOnCl 𝕜 (f ∘ g) t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {s : Set E} {g : G → E} {t : Set G} (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g t) (h : Set.MapsTo g t s)
Goal: DiffContOnCl 𝕜 (f ∘ g) t
-- DiffContOnCl.continuousOn_ball
theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
    ContinuousOn f (closedBall x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (Metric.ball x r))
Goal: ContinuousOn f (Metric.closedBall x r)
-- DiffContOnCl.mk_ball
theorem mk_ball {x : E} {r : ℝ} (hd : DifferentiableOn 𝕜 f (ball x r))
    (hc : ContinuousOn f (closedBall x r)) : DiffContOnCl 𝕜 f (ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {x : E} {r : ℝ} (hd : DifferentiableOn 𝕜 f (Metric.ball x r)) (hc : ContinuousOn f (Metric.closedBall x r))
Goal: DiffContOnCl 𝕜 f (Metric.ball x r)
-- DiffContOnCl.differentiableAt
protected theorem differentiableAt (h : DiffContOnCl 𝕜 f s) (hs : IsOpen s) (hx : x ∈ s) :
    DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : DiffContOnCl 𝕜 f s) (hs : IsOpen s) (hx : x ∈ s)
Goal: DifferentiableAt 𝕜 f x
-- DiffContOnCl.differentiableAt'
theorem differentiableAt' (h : DiffContOnCl 𝕜 f s) (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {x : E} (h : DiffContOnCl 𝕜 f s) (hx : s ∈ 𝓝 x)
Goal: DifferentiableAt 𝕜 f x
-- DiffContOnCl.mono
protected theorem mono (h : DiffContOnCl 𝕜 f s) (ht : t ⊆ s) : DiffContOnCl 𝕜 f t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s t : Set E} (h : DiffContOnCl 𝕜 f s) (ht : t ⊆ s)
Goal: DiffContOnCl 𝕜 f t
-- DiffContOnCl.add
theorem add (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s) : DiffContOnCl 𝕜 (f + g) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s)
Goal: DiffContOnCl 𝕜 (f + g) s
-- DiffContOnCl.add_const
theorem add_const (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => f x + c) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (c : F)
Goal: DiffContOnCl 𝕜 (fun x => f x + c) s
-- DiffContOnCl.const_add
theorem const_add (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => c + f x) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (c : F)
Goal: DiffContOnCl 𝕜 (fun x => c + f x) s
-- DiffContOnCl.neg
theorem neg (hf : DiffContOnCl 𝕜 f s) : DiffContOnCl 𝕜 (-f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s)
Goal: DiffContOnCl 𝕜 (-f) s
-- DiffContOnCl.sub
theorem sub (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s) : DiffContOnCl 𝕜 (f - g) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (hg : DiffContOnCl 𝕜 g s)
Goal: DiffContOnCl 𝕜 (f - g) s
-- DiffContOnCl.sub_const
theorem sub_const (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => f x - c) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (c : F)
Goal: DiffContOnCl 𝕜 (fun x => f x - c) s
-- DiffContOnCl.const_sub
theorem const_sub (hf : DiffContOnCl 𝕜 f s) (c : F) : DiffContOnCl 𝕜 (fun x => c - f x) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : DiffContOnCl 𝕜 f s) (c : F)
Goal: DiffContOnCl 𝕜 (fun x => c - f x) s
-- DiffContOnCl.const_smul
theorem const_smul {R : Type*} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F]
    [ContinuousConstSMul R F] (hf : DiffContOnCl 𝕜 f s) (c : R) : DiffContOnCl 𝕜 (c • f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} {R : Type u_5} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (hf : DiffContOnCl 𝕜 f s) (c : R)
Goal: DiffContOnCl 𝕜 (c • f) s
-- DiffContOnCl.smul
theorem smul {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F]
    [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {f : E → F} {s : Set E} (hc : DiffContOnCl 𝕜 c s)
    (hf : DiffContOnCl 𝕜 f s) : DiffContOnCl 𝕜 (fun x => c x • f x) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {𝕜' : Type u_5} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {f : E → F} {s : Set E} (hc : DiffContOnCl 𝕜 c s) (hf : DiffContOnCl 𝕜 f s)
Goal: DiffContOnCl 𝕜 (fun x => c x • f x) s
-- DiffContOnCl.smul_const
theorem smul_const {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜']
    [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {s : Set E} (hc : DiffContOnCl 𝕜 c s)
    (y : F) : DiffContOnCl 𝕜 (fun x => c x • y) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {𝕜' : Type u_5} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : E → 𝕜'} {s : Set E} (hc : DiffContOnCl 𝕜 c s) (y : F)
Goal: DiffContOnCl 𝕜 (fun x => c x • y) s
-- DiffContOnCl.inv
theorem inv {f : E → 𝕜} (hf : DiffContOnCl 𝕜 f s) (h₀ : ∀ x ∈ closure s, f x ≠ 0) :
    DiffContOnCl 𝕜 f⁻¹ s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E} {f : E → 𝕜} (hf : DiffContOnCl 𝕜 f s) (h₀ : ∀ x ∈ closure s, f x ≠ 0)
Goal: DiffContOnCl 𝕜 f⁻¹ s
-- Differentiable.comp_diffContOnCl
theorem Differentiable.comp_diffContOnCl {g : G → E} {t : Set G} (hf : Differentiable 𝕜 f)
    (hg : DiffContOnCl 𝕜 g t) : DiffContOnCl 𝕜 (f ∘ g) t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {g : G → E} {t : Set G} (hf : Differentiable 𝕜 f) (hg : DiffContOnCl 𝕜 g t)
Goal: DiffContOnCl 𝕜 (f ∘ g) t
-- DifferentiableOn.diffContOnCl_ball
theorem DifferentiableOn.diffContOnCl_ball {U : Set E} {c : E} {R : ℝ} (hf : DifferentiableOn 𝕜 f U)
    (hc : closedBall c R ⊆ U) : DiffContOnCl 𝕜 f (ball c R) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {U : Set E} {c : E} {R : ℝ} (hf : DifferentiableOn 𝕜 f U) (hc : Metric.closedBall c R ⊆ U)
Goal: DiffContOnCl 𝕜 f (Metric.ball c R)
-- HasDerivAt.lhopital_zero_right_on_Ioo
theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
    (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
    (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
    Tendsto (fun x => f x / g x) (𝓝[>] a) l 
Arguments: {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ} (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Set.Ioo a b, g' x ≠ 0) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 0)) (hdiv : Filter.Tendsto (fun x => f' x / g' x) (𝓝[>] a) l)
Goal: Filter.Tendsto (fun x => f x / g x) (𝓝[>] a) l