ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt
theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
    (K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, LipschitzOnWith 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} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x) (K : ℝ≥0) (hK : ‖f'‖₊ < K)
Goal: ∃ s ∈ 𝓝 x, LipschitzOnWith K f s
-- HasStrictFDerivAt.exists_lipschitzOnWith
theorem HasStrictFDerivAt.exists_lipschitzOnWith (hf : HasStrictFDerivAt f f' x) :
    ∃ K, ∃ s ∈ 𝓝 x, LipschitzOnWith 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} {f' : E →L[𝕜] F} {x : E} (hf : HasStrictFDerivAt f f' x)
Goal: ∃ K, ∃ s ∈ 𝓝 x, LipschitzOnWith K f s
-- HasFDerivAt.lim
theorem HasFDerivAt.lim (hf : HasFDerivAt f f' x) (v : E) {α : Type*} {c : α → 𝕜} {l : Filter α}
    (hc : Tendsto (fun n => ‖c n‖) l atTop) :
    Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' 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] {f : E → F} {f' : E →L[𝕜] F} {x : E} (hf : HasFDerivAt f f' x) (v : E) {α : Type u_6} {c : α → 𝕜} {l : Filter α} (hc : Filter.Tendsto (fun n => ‖c n‖) l Filter.atTop)
Goal: Filter.Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v))
-- HasFDerivAt.unique
theorem HasFDerivAt.unique (h₀ : HasFDerivAt f f₀' x) (h₁ : HasFDerivAt f f₁' x) : f₀' = f₁' 
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} {f₀' f₁' : E →L[𝕜] F} {x : E} (h₀ : HasFDerivAt f f₀' x) (h₁ : HasFDerivAt f f₁' x)
Goal: f₀' = f₁'
-- hasFDerivWithinAt_inter'
theorem hasFDerivWithinAt_inter' (h : t ∈ 𝓝[s] x) :
    HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f 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 t : Set E} (h : t ∈ 𝓝[s] x)
Goal: HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x
-- hasFDerivWithinAt_inter
theorem hasFDerivWithinAt_inter (h : t ∈ 𝓝 x) :
    HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f 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 t : Set E} (h : t ∈ 𝓝 x)
Goal: HasFDerivWithinAt f f' (s ∩ t) x ↔ HasFDerivWithinAt f f' s x
-- HasFDerivWithinAt.union
theorem HasFDerivWithinAt.union (hs : HasFDerivWithinAt f f' s x)
    (ht : HasFDerivWithinAt f f' t x) : HasFDerivWithinAt f f' (s ∪ 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] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (hs : HasFDerivWithinAt f f' s x) (ht : HasFDerivWithinAt f f' t x)
Goal: HasFDerivWithinAt f f' (s ∪ t) x
-- HasFDerivWithinAt.hasFDerivAt
theorem HasFDerivWithinAt.hasFDerivAt (h : HasFDerivWithinAt f f' s x) (hs : s ∈ 𝓝 x) :
    HasFDerivAt f 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} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hs : s ∈ 𝓝 x)
Goal: HasFDerivAt f f' x
-- DifferentiableWithinAt.differentiableAt
theorem DifferentiableWithinAt.differentiableAt (h : DifferentiableWithinAt 𝕜 f s x)
    (hs : s ∈ 𝓝 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} {s : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (hs : s ∈ 𝓝 x)
Goal: DifferentiableAt 𝕜 f x
-- HasFDerivWithinAt.of_nhdsWithin_eq_bot
theorem HasFDerivWithinAt.of_nhdsWithin_eq_bot (h : 𝓝[s\{x}] x = ⊥) :
    HasFDerivWithinAt f 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 : 𝓝[s \ {x}] x = ⊥)
Goal: HasFDerivWithinAt f f' s x
-- hasFDerivWithinAt_of_nmem_closure
theorem hasFDerivWithinAt_of_nmem_closure (h : x ∉ closure s) : HasFDerivWithinAt f 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 : x ∉ closure s)
Goal: HasFDerivWithinAt f f' s x
-- DifferentiableWithinAt.hasFDerivWithinAt
theorem DifferentiableWithinAt.hasFDerivWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    HasFDerivWithinAt f (fderivWithin 𝕜 f s x) 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: HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x
-- DifferentiableAt.hasFDerivAt
theorem DifferentiableAt.hasFDerivAt (h : DifferentiableAt 𝕜 f x) :
    HasFDerivAt f (fderiv 𝕜 f 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] {f : E → F} {x : E} (h : DifferentiableAt 𝕜 f x)
Goal: HasFDerivAt f (fderiv 𝕜 f x) x
-- DifferentiableOn.hasFDerivAt
theorem DifferentiableOn.hasFDerivAt (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x) :
    HasFDerivAt f (fderiv 𝕜 f 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] {f : E → F} {x : E} {s : Set E} (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x)
Goal: HasFDerivAt f (fderiv 𝕜 f x) x
-- DifferentiableOn.differentiableAt
theorem DifferentiableOn.differentiableAt (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 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} {s : Set E} (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x)
Goal: DifferentiableAt 𝕜 f x
-- DifferentiableOn.eventually_differentiableAt
theorem DifferentiableOn.eventually_differentiableAt (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x) :
    ∀ᶠ y in 𝓝 x, DifferentiableAt 𝕜 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} {x : E} {s : Set E} (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x)
Goal: ∀ᶠ (y : E) in 𝓝 x, DifferentiableAt 𝕜 f y
-- HasFDerivAt.fderiv
protected theorem HasFDerivAt.fderiv (h : HasFDerivAt f f' x) : fderiv 𝕜 f x = 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} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x)
Goal: fderiv 𝕜 f x = f'
-- fderiv_eq
theorem fderiv_eq {f' : E → E →L[𝕜] F} (h : ∀ x, HasFDerivAt f (f' x) x) : fderiv 𝕜 f = 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} {f' : E → E →L[𝕜] F} (h : ∀ (x : E), HasFDerivAt f (f' x) x)
Goal: fderiv 𝕜 f = f'
-- norm_fderiv_le_of_lip'
theorem norm_fderiv_le_of_lip' {f : E → F} {x₀ : E}
    {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
    ‖fderiv 𝕜 f x₀‖ ≤ 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} {x₀ : E} {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖)
Goal: ‖fderiv 𝕜 f x₀‖ ≤ C
-- norm_fderiv_le_of_lipschitzOn
-- Porting note: renamed so that dot-notation makes sense
theorem norm_fderiv_le_of_lipschitzOn {f : E → F} {x₀ : E} {s : Set E} (hs : s ∈ 𝓝 x₀)
    {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖fderiv 𝕜 f x₀‖ ≤ 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} {x₀ : E} {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖fderiv 𝕜 f x₀‖ ≤ ↑C
-- norm_fderiv_le_of_lipschitz
theorem norm_fderiv_le_of_lipschitz {f : E → F} {x₀ : E}
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖fderiv 𝕜 f x₀‖ ≤ 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} {x₀ : E} {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖fderiv 𝕜 f x₀‖ ≤ ↑C
-- HasFDerivWithinAt.fderivWithin
protected theorem HasFDerivWithinAt.fderivWithin (h : HasFDerivWithinAt f f' s x)
    (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 f s x = 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} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 f s x = f'
-- DifferentiableWithinAt.mono
theorem DifferentiableWithinAt.mono (h : DifferentiableWithinAt 𝕜 f t x) (st : s ⊆ t) :
    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 t : Set E} (h : DifferentiableWithinAt 𝕜 f t x) (st : s ⊆ t)
Goal: DifferentiableWithinAt 𝕜 f s x
-- DifferentiableWithinAt.mono_of_mem
theorem DifferentiableWithinAt.mono_of_mem (h : DifferentiableWithinAt 𝕜 f s x) {t : Set E}
    (hst : s ∈ 𝓝[t] x) : DifferentiableWithinAt 𝕜 f 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] {f : E → F} {x : E} {s : Set E} (h : DifferentiableWithinAt 𝕜 f s x) {t : Set E} (hst : s ∈ 𝓝[t] x)
Goal: DifferentiableWithinAt 𝕜 f t x
-- differentiableWithinAt_univ
theorem differentiableWithinAt_univ :
    DifferentiableWithinAt 𝕜 f univ 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: DifferentiableWithinAt 𝕜 f Set.univ x ↔ DifferentiableAt 𝕜 f x
-- differentiableWithinAt_inter
theorem differentiableWithinAt_inter (ht : t ∈ 𝓝 x) :
    DifferentiableWithinAt 𝕜 f (s ∩ t) 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 t : Set E} (ht : t ∈ 𝓝 x)
Goal: DifferentiableWithinAt 𝕜 f (s ∩ t) x ↔ DifferentiableWithinAt 𝕜 f s x
-- differentiableWithinAt_inter'
theorem differentiableWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
    DifferentiableWithinAt 𝕜 f (s ∩ t) 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 t : Set E} (ht : t ∈ 𝓝[s] x)
Goal: DifferentiableWithinAt 𝕜 f (s ∩ t) x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableAt.differentiableWithinAt
theorem DifferentiableAt.differentiableWithinAt (h : DifferentiableAt 𝕜 f 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} (h : DifferentiableAt 𝕜 f x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- Differentiable.differentiableAt
@[fun_prop]
theorem Differentiable.differentiableAt (h : Differentiable 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 : Differentiable 𝕜 f)
Goal: DifferentiableAt 𝕜 f x
-- DifferentiableAt.fderivWithin
protected theorem DifferentiableAt.fderivWithin (h : DifferentiableAt 𝕜 f x)
    (hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 f s 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} {s : Set E} (h : DifferentiableAt 𝕜 f x) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 f s x = fderiv 𝕜 f x
-- DifferentiableOn.mono
theorem DifferentiableOn.mono (h : DifferentiableOn 𝕜 f t) (st : s ⊆ t) : 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 t : Set E} (h : DifferentiableOn 𝕜 f t) (st : s ⊆ t)
Goal: DifferentiableOn 𝕜 f s
-- differentiableOn_univ
theorem differentiableOn_univ : DifferentiableOn 𝕜 f univ ↔ 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: DifferentiableOn 𝕜 f Set.univ ↔ Differentiable 𝕜 f
-- Differentiable.differentiableOn
@[fun_prop]
theorem Differentiable.differentiableOn (h : Differentiable 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 : Differentiable 𝕜 f)
Goal: DifferentiableOn 𝕜 f s
-- differentiableOn_of_locally_differentiableOn
theorem differentiableOn_of_locally_differentiableOn
    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ DifferentiableOn 𝕜 f (s ∩ u)) :
    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 : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ DifferentiableOn 𝕜 f (s ∩ u))
Goal: DifferentiableOn 𝕜 f s
-- fderivWithin_of_mem
theorem fderivWithin_of_mem (st : t ∈ 𝓝[s] x) (ht : UniqueDiffWithinAt 𝕜 s x)
    (h : DifferentiableWithinAt 𝕜 f t x) : fderivWithin 𝕜 f s x = fderivWithin 𝕜 f 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] {f : E → F} {x : E} {s t : Set E} (st : t ∈ 𝓝[s] x) (ht : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f t x)
Goal: fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x
-- fderivWithin_subset
theorem fderivWithin_subset (st : s ⊆ t) (ht : UniqueDiffWithinAt 𝕜 s x)
    (h : DifferentiableWithinAt 𝕜 f t x) : fderivWithin 𝕜 f s x = fderivWithin 𝕜 f 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] {f : E → F} {x : E} {s t : Set E} (st : s ⊆ t) (ht : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f t x)
Goal: fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x
-- fderivWithin_inter
theorem fderivWithin_inter (ht : t ∈ 𝓝 x) : fderivWithin 𝕜 f (s ∩ t) 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 t : Set E} (ht : t ∈ 𝓝 x)
Goal: fderivWithin 𝕜 f (s ∩ t) x = fderivWithin 𝕜 f s x
-- fderivWithin_univ
@[simp]
theorem fderivWithin_univ : fderivWithin 𝕜 f univ = fderiv 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: fderivWithin 𝕜 f Set.univ = fderiv 𝕜 f
-- fderivWithin_of_mem_nhds
theorem fderivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : fderivWithin 𝕜 f s 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} {s : Set E} (h : s ∈ 𝓝 x)
Goal: fderivWithin 𝕜 f s x = fderiv 𝕜 f x
-- fderivWithin_of_isOpen
theorem fderivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) : fderivWithin 𝕜 f s 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} {s : Set E} (hs : IsOpen s) (hx : x ∈ s)
Goal: fderivWithin 𝕜 f s x = fderiv 𝕜 f x
-- fderivWithin_eq_fderiv
theorem fderivWithin_eq_fderiv (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableAt 𝕜 f x) :
    fderivWithin 𝕜 f s 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} {s : Set E} (hs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableAt 𝕜 f x)
Goal: fderivWithin 𝕜 f s x = fderiv 𝕜 f x
-- fderiv_mem_iff
theorem fderiv_mem_iff {f : E → F} {s : Set (E →L[𝕜] F)} {x : E} : fderiv 𝕜 f x ∈ s ↔
    DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ (0 : E →L[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 →L[𝕜] F)} {x : E}
Goal: fderiv 𝕜 f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s
-- fderivWithin_mem_iff
theorem fderivWithin_mem_iff {f : E → F} {t : Set E} {s : Set (E →L[𝕜] F)} {x : E} :
    fderivWithin 𝕜 f t x ∈ s ↔
      DifferentiableWithinAt 𝕜 f t x ∧ fderivWithin 𝕜 f t x ∈ s ∨
        ¬DifferentiableWithinAt 𝕜 f t x ∧ (0 : E →L[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} {t : Set E} {s : Set (E →L[𝕜] F)} {x : E}
Goal: fderivWithin 𝕜 f t x ∈ s ↔
  DifferentiableWithinAt 𝕜 f t x ∧ fderivWithin 𝕜 f t x ∈ s ∨ ¬DifferentiableWithinAt 𝕜 f t x ∧ 0 ∈ s
-- Asymptotics.IsBigO.hasFDerivWithinAt
theorem Asymptotics.IsBigO.hasFDerivWithinAt {s : Set E} {x₀ : E} {n : ℕ}
    (h : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n) (hx₀ : x₀ ∈ s) (hn : 1 < n) :
    HasFDerivWithinAt f (0 : E →L[𝕜] F) s x₀ 
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} {x₀ : E} {n : ℕ} (h : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n) (hx₀ : x₀ ∈ s) (hn : 1 < n)
Goal: HasFDerivWithinAt f 0 s x₀
-- Asymptotics.IsBigO.hasFDerivAt
theorem Asymptotics.IsBigO.hasFDerivAt {x₀ : E} {n : ℕ} (h : f =O[𝓝 x₀] fun x => ‖x - x₀‖ ^ n)
    (hn : 1 < n) : HasFDerivAt f (0 : E →L[𝕜] F) x₀ 
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} {x₀ : E} {n : ℕ} (h : f =O[𝓝 x₀] fun x => ‖x - x₀‖ ^ n) (hn : 1 < n)
Goal: HasFDerivAt f 0 x₀
-- HasFDerivWithinAt.isBigO_sub
nonrec theorem HasFDerivWithinAt.isBigO_sub {f : E → F} {s : Set E} {x₀ : E} {f' : E →L[𝕜] F}
    (h : HasFDerivWithinAt f f' s x₀) : (f · - f x₀) =O[𝓝[s] x₀] (· - x₀) 
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} {x₀ : E} {f' : E →L[𝕜] F} (h : HasFDerivWithinAt f f' s x₀)
Goal: (fun x => f x - f x₀) =O[𝓝[s] x₀] fun x => x - x₀
-- DifferentiableWithinAt.isBigO_sub
lemma DifferentiableWithinAt.isBigO_sub {f : E → F} {s : Set E} {x₀ : E}
    (h : DifferentiableWithinAt 𝕜 f s x₀) : (f · - f x₀) =O[𝓝[s] x₀] (· - x₀) 
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} {x₀ : E} (h : DifferentiableWithinAt 𝕜 f s x₀)
Goal: (fun x => f x - f x₀) =O[𝓝[s] x₀] fun x => x - x₀
-- HasFDerivAt.isBigO_sub
nonrec theorem HasFDerivAt.isBigO_sub {f : E → F} {x₀ : E} {f' : E →L[𝕜] F}
    (h : HasFDerivAt f f' x₀) : (f · - f x₀) =O[𝓝 x₀] (· - x₀) 
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} {x₀ : E} {f' : E →L[𝕜] F} (h : HasFDerivAt f f' x₀)
Goal: (fun x => f x - f x₀) =O[𝓝 x₀] fun x => x - x₀
-- DifferentiableAt.isBigO_sub
nonrec theorem DifferentiableAt.isBigO_sub {f : E → F} {x₀ : E} (h : DifferentiableAt 𝕜 f x₀) :
    (f · - f x₀) =O[𝓝 x₀] (· - x₀) 
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} {x₀ : E} (h : DifferentiableAt 𝕜 f x₀)
Goal: (fun x => f x - f x₀) =O[𝓝 x₀] fun x => x - x₀
-- HasFDerivAtFilter.tendsto_nhds
theorem HasFDerivAtFilter.tendsto_nhds (hL : L ≤ 𝓝 x) (h : HasFDerivAtFilter f f' x L) :
    Tendsto f L (𝓝 (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} {L : Filter E} (hL : L ≤ 𝓝 x) (h : HasFDerivAtFilter f f' x L)
Goal: Filter.Tendsto f L (𝓝 (f x))
-- HasFDerivWithinAt.continuousWithinAt
theorem HasFDerivWithinAt.continuousWithinAt (h : HasFDerivWithinAt f f' s x) :
    ContinuousWithinAt 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: ContinuousWithinAt f s x
-- HasFDerivAt.continuousAt
theorem HasFDerivAt.continuousAt (h : HasFDerivAt f f' x) : ContinuousAt 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: ContinuousAt f x
-- DifferentiableWithinAt.continuousWithinAt
@[fun_prop]
theorem DifferentiableWithinAt.continuousWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    ContinuousWithinAt 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 : DifferentiableWithinAt 𝕜 f s x)
Goal: ContinuousWithinAt f s x
-- DifferentiableAt.continuousAt
@[fun_prop]
theorem DifferentiableAt.continuousAt (h : DifferentiableAt 𝕜 f x) : ContinuousAt 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 : DifferentiableAt 𝕜 f x)
Goal: ContinuousAt f x
-- DifferentiableOn.continuousOn
@[fun_prop]
theorem DifferentiableOn.continuousOn (h : DifferentiableOn 𝕜 f s) : ContinuousOn 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 : DifferentiableOn 𝕜 f s)
Goal: ContinuousOn f s