ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hasFDerivAt_iff_hasGradientAt
theorem hasFDerivAt_iff_hasGradientAt {frechet : F →L[𝕜] 𝕜} :
    HasFDerivAt f frechet x ↔ HasGradientAt f ((toDual 𝕜 F).symm frechet) x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {frechet : F →L[𝕜] 𝕜}
Goal: HasFDerivAt f frechet x ↔ HasGradientAt f ((InnerProductSpace.toDual 𝕜 F).symm frechet) x
-- gradient_eq_zero_of_not_differentiableAt
theorem gradient_eq_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : ∇ f x = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} (h : ¬DifferentiableAt 𝕜 f x)
Goal: ∇ f x = 0
-- HasGradientAt.unique
theorem HasGradientAt.unique {gradf gradg : F}
    (hf : HasGradientAt f gradf x) (hg : HasGradientAt f gradg x) :
    gradf = gradg 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x gradf gradg : F} (hf : HasGradientAt f gradf x) (hg : HasGradientAt f gradg x)
Goal: gradf = gradg
-- DifferentiableAt.hasGradientAt
theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
    HasGradientAt f (∇ f x) x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} (h : DifferentiableAt 𝕜 f x)
Goal: HasGradientAt f (∇ f x) x
-- HasGradientAt.differentiableAt
theorem HasGradientAt.differentiableAt (h : HasGradientAt f f' x) :
    DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} (h : HasGradientAt f f' x)
Goal: DifferentiableAt 𝕜 f x
-- DifferentiableWithinAt.hasGradientWithinAt
theorem DifferentiableWithinAt.hasGradientWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
    HasGradientWithinAt f (gradientWithin f s x) s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {s : Set F} (h : DifferentiableWithinAt 𝕜 f s x)
Goal: HasGradientWithinAt f (gradientWithin f s x) s x
-- HasGradientWithinAt.differentiableWithinAt
theorem HasGradientWithinAt.differentiableWithinAt (h : HasGradientWithinAt f f' s x) :
    DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} (h : HasGradientWithinAt f f' s x)
Goal: DifferentiableWithinAt 𝕜 f s x
-- hasGradientWithinAt_univ
@[simp]
theorem hasGradientWithinAt_univ : HasGradientWithinAt f f' univ x ↔ HasGradientAt f f' x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F}
Goal: HasGradientWithinAt f f' Set.univ x ↔ HasGradientAt f f' x
-- DifferentiableOn.hasGradientAt
theorem DifferentiableOn.hasGradientAt (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x) :
    HasGradientAt f (∇ f x) x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {s : Set F} (h : DifferentiableOn 𝕜 f s) (hs : s ∈ 𝓝 x)
Goal: HasGradientAt f (∇ f x) x
-- HasGradientAt.gradient
theorem HasGradientAt.gradient (h : HasGradientAt f f' x) : ∇ f x = f' 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} (h : HasGradientAt f f' x)
Goal: ∇ f x = f'
-- gradient_eq
theorem gradient_eq {f' : F → F} (h : ∀ x, HasGradientAt f (f' x) x) : ∇ f = f' 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' : F → F} (h : ∀ (x : F), HasGradientAt f (f' x) x)
Goal: ∇ f = f'
-- HasGradientAtFilter.hasDerivAtFilter
theorem HasGradientAtFilter.hasDerivAtFilter (h : HasGradientAtFilter g g' u L') :
    HasDerivAtFilter g (starRingEnd 𝕜 g') u L' 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {g : 𝕜 → 𝕜} {g' u : 𝕜} {L' : Filter 𝕜} (h : HasGradientAtFilter g g' u L')
Goal: HasDerivAtFilter g ((starRingEnd 𝕜) g') u L'
-- HasDerivAtFilter.hasGradientAtFilter
theorem HasDerivAtFilter.hasGradientAtFilter (h : HasDerivAtFilter g g' u L') :
    HasGradientAtFilter g (starRingEnd 𝕜 g') u L' 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {g : 𝕜 → 𝕜} {g' u : 𝕜} {L' : Filter 𝕜} (h : HasDerivAtFilter g g' u L')
Goal: HasGradientAtFilter g ((starRingEnd 𝕜) g') u L'
-- HasGradientAt.hasDerivAt
theorem HasGradientAt.hasDerivAt (h : HasGradientAt g g' u) :
    HasDerivAt g (starRingEnd 𝕜 g') u 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {g : 𝕜 → 𝕜} {g' u : 𝕜} (h : HasGradientAt g g' u)
Goal: HasDerivAt g ((starRingEnd 𝕜) g') u
-- HasDerivAt.hasGradientAt
theorem HasDerivAt.hasGradientAt (h : HasDerivAt g g' u) :
    HasGradientAt g (starRingEnd 𝕜 g') u 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {g : 𝕜 → 𝕜} {g' u : 𝕜} (h : HasDerivAt g g' u)
Goal: HasGradientAt g ((starRingEnd 𝕜) g') u
-- gradient_eq_deriv
theorem gradient_eq_deriv : ∇ g u = starRingEnd 𝕜 (deriv g u) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {g : 𝕜 → 𝕜} {u : 𝕜}
Goal: ∇ g u = (starRingEnd 𝕜) (deriv g u)
-- HasGradientAtFilter.hasDerivAtFilter'
theorem HasGradientAtFilter.hasDerivAtFilter' (h : HasGradientAtFilter g g' u L') :
    HasDerivAtFilter g g' u L' 
Arguments: {g : ℝ → ℝ} {g' u : ℝ} {L' : Filter ℝ} (h : HasGradientAtFilter g g' u L')
Goal: HasDerivAtFilter g g' u L'
-- HasDerivAtFilter.hasGradientAtFilter'
theorem HasDerivAtFilter.hasGradientAtFilter' (h : HasDerivAtFilter g g' u L') :
    HasGradientAtFilter g g' u L' 
Arguments: {g : ℝ → ℝ} {g' u : ℝ} {L' : Filter ℝ} (h : HasDerivAtFilter g g' u L')
Goal: HasGradientAtFilter g g' u L'
-- HasGradientAt.hasDerivAt'
theorem HasGradientAt.hasDerivAt' (h : HasGradientAt g g' u) :
    HasDerivAt g g' u 
Arguments: {g : ℝ → ℝ} {g' u : ℝ} (h : HasGradientAt g g' u)
Goal: HasDerivAt g g' u
-- HasDerivAt.hasGradientAt'
theorem HasDerivAt.hasGradientAt' (h : HasDerivAt g g' u) :
    HasGradientAt g g' u 
Arguments: {g : ℝ → ℝ} {g' u : ℝ} (h : HasDerivAt g g' u)
Goal: HasGradientAt g g' u
-- gradient_eq_deriv'
theorem gradient_eq_deriv' : ∇ g u = deriv g u 
Arguments: {g : ℝ → ℝ} {u : ℝ}
Goal: ∇ g u = deriv g u
-- hasGradientAtFilter_iff_isLittleO
theorem hasGradientAtFilter_iff_isLittleO :
    HasGradientAtFilter f f' x L ↔
    (fun x' : F => f x' - f x - ⟪f', x' - x⟫) =o[L] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {L : Filter F}
Goal: HasGradientAtFilter f f' x L ↔ (fun x' => f x' - f x - ⟪f', x' - x⟫_𝕜) =o[L] fun x' => x' - x
-- hasGradientWithinAt_iff_isLittleO
theorem hasGradientWithinAt_iff_isLittleO :
    HasGradientWithinAt f f' s x ↔
    (fun x' : F => f x' - f x - ⟪f', x' - x⟫) =o[𝓝[s] x] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F}
Goal: HasGradientWithinAt f f' s x ↔ (fun x' => f x' - f x - ⟪f', x' - x⟫_𝕜) =o[𝓝[s] x] fun x' => x' - x
-- hasGradientWithinAt_iff_tendsto
theorem hasGradientWithinAt_iff_tendsto :
    HasGradientWithinAt f f' s x ↔
    Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - ⟪f', x' - x⟫‖) (𝓝[s] x) (𝓝 0) 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F}
Goal: HasGradientWithinAt f f' s x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - ⟪f', x' - x⟫_𝕜‖) (𝓝[s] x) (𝓝 0)
-- hasGradientAt_iff_isLittleO
theorem hasGradientAt_iff_isLittleO : HasGradientAt f f' x ↔
    (fun x' : F => f x' - f x - ⟪f', x' - x⟫) =o[𝓝 x] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F}
Goal: HasGradientAt f f' x ↔ (fun x' => f x' - f x - ⟪f', x' - x⟫_𝕜) =o[𝓝 x] fun x' => x' - x
-- hasGradientAt_iff_tendsto
theorem hasGradientAt_iff_tendsto :
    HasGradientAt f f' x ↔
    Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - ⟪f', x' - x⟫‖) (𝓝 x) (𝓝 0) 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F}
Goal: HasGradientAt f f' x ↔ Filter.Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - ⟪f', x' - x⟫_𝕜‖) (𝓝 x) (𝓝 0)
-- HasGradientAtFilter.isBigO_sub
theorem HasGradientAtFilter.isBigO_sub (h : HasGradientAtFilter f f' x L) :
    (fun x' => f x' - f x) =O[L] fun x' => x' - x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {L : Filter F} (h : HasGradientAtFilter f f' x L)
Goal: (fun x' => f x' - f x) =O[L] fun x' => x' - x
-- hasGradientWithinAt_congr_set'
theorem hasGradientWithinAt_congr_set' {s t : Set F} (y : F) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    HasGradientWithinAt f f' s x ↔ HasGradientWithinAt f f' t x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s t : Set F} (y : F) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: HasGradientWithinAt f f' s x ↔ HasGradientWithinAt f f' t x
-- hasGradientWithinAt_congr_set
theorem hasGradientWithinAt_congr_set {s t : Set F} (h : s =ᶠ[𝓝 x] t) :
    HasGradientWithinAt f f' s x ↔ HasGradientWithinAt f f' t x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s t : Set F} (h : s =ᶠ[𝓝 x] t)
Goal: HasGradientWithinAt f f' s x ↔ HasGradientWithinAt f f' t x
-- hasGradientAt_iff_isLittleO_nhds_zero
theorem hasGradientAt_iff_isLittleO_nhds_zero : HasGradientAt f f' x ↔
    (fun h => f (x + h) - f x - ⟪f', h⟫) =o[𝓝 0] fun h => h 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F}
Goal: HasGradientAt f f' x ↔ (fun h => f (x + h) - f x - ⟪f', h⟫_𝕜) =o[𝓝 0] fun h => h
-- Filter.EventuallyEq.hasGradientAtFilter_iff
theorem Filter.EventuallyEq.hasGradientAtFilter_iff (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x)
    (h₁ : f₀' = f₁') : HasGradientAtFilter f₀ f₀' x L ↔ HasGradientAtFilter f₁ f₁' x L 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {x : F} {L : Filter F} {f₀ f₁ : F → 𝕜} {f₀' f₁' : F} (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : f₀' = f₁')
Goal: HasGradientAtFilter f₀ f₀' x L ↔ HasGradientAtFilter f₁ f₁' x L
-- HasGradientAtFilter.congr_of_eventuallyEq
theorem HasGradientAtFilter.congr_of_eventuallyEq (h : HasGradientAtFilter f f' x L)
    (hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : HasGradientAtFilter f₁ f' x L 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {L : Filter F} {f₁ : F → 𝕜} (h : HasGradientAtFilter f f' x L) (hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x)
Goal: HasGradientAtFilter f₁ f' x L
-- HasGradientWithinAt.congr_mono
theorem HasGradientWithinAt.congr_mono (h : HasGradientWithinAt f f' s x) (ht : ∀ x ∈ t, f₁ x = f x)
    (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasGradientWithinAt f₁ f' t x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} {f₁ : F → 𝕜} {t : Set F} (h : HasGradientWithinAt f f' s x) (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: HasGradientWithinAt f₁ f' t x
-- HasGradientWithinAt.congr
theorem HasGradientWithinAt.congr (h : HasGradientWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x)
    (hx : f₁ x = f x) : HasGradientWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} {f₁ : F → 𝕜} (h : HasGradientWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: HasGradientWithinAt f₁ f' s x
-- HasGradientWithinAt.congr_of_mem
theorem HasGradientWithinAt.congr_of_mem (h : HasGradientWithinAt f f' s x)
    (hs : ∀ x ∈ s, f₁ x = f x) (hx : x ∈ s) : HasGradientWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} {f₁ : F → 𝕜} (h : HasGradientWithinAt f f' s x) (hs : ∀ x ∈ s, f₁ x = f x) (hx : x ∈ s)
Goal: HasGradientWithinAt f₁ f' s x
-- HasGradientWithinAt.congr_of_eventuallyEq
theorem HasGradientWithinAt.congr_of_eventuallyEq (h : HasGradientWithinAt f f' s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasGradientWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} {f₁ : F → 𝕜} (h : HasGradientWithinAt f f' s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasGradientWithinAt f₁ f' s x
-- HasGradientWithinAt.congr_of_eventuallyEq_of_mem
theorem HasGradientWithinAt.congr_of_eventuallyEq_of_mem (h : HasGradientWithinAt f f' s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) : HasGradientWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} {f₁ : F → 𝕜} (h : HasGradientWithinAt f f' s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s)
Goal: HasGradientWithinAt f₁ f' s x
-- HasGradientAt.congr_of_eventuallyEq
theorem HasGradientAt.congr_of_eventuallyEq (h : HasGradientAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasGradientAt f₁ f' x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {f₁ : F → 𝕜} (h : HasGradientAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f)
Goal: HasGradientAt f₁ f' x
-- Filter.EventuallyEq.gradient_eq
theorem Filter.EventuallyEq.gradient_eq (hL : f₁ =ᶠ[𝓝 x] f) : ∇ f₁ x = ∇ f x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {f₁ : F → 𝕜} (hL : f₁ =ᶠ[𝓝 x] f)
Goal: ∇ f₁ x = ∇ f x
-- Filter.EventuallyEq.gradient
protected theorem Filter.EventuallyEq.gradient (h : f₁ =ᶠ[𝓝 x] f) : ∇ f₁ =ᶠ[𝓝 x] ∇ f 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {x : F} {f₁ : F → 𝕜} (h : f₁ =ᶠ[𝓝 x] f)
Goal: ∇ f₁ =ᶠ[𝓝 x] ∇ f
-- hasGradientAtFilter_const
theorem hasGradientAtFilter_const : HasGradientAtFilter (fun _ => c) 0 x L 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (x : F) (L : Filter F) (c : 𝕜)
Goal: HasGradientAtFilter (fun x => c) 0 x L
-- hasGradientWithinAt_const
theorem hasGradientWithinAt_const : HasGradientWithinAt (fun _ => c) 0 s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (x : F) (s : Set F) (c : 𝕜)
Goal: HasGradientWithinAt (fun x => c) 0 s x
-- hasGradientAt_const
theorem hasGradientAt_const : HasGradientAt (fun _ => c) 0 x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (x : F) (c : 𝕜)
Goal: HasGradientAt (fun x => c) 0 x
-- gradient_const
theorem gradient_const : ∇ (fun _ => c) x = 0 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (x : F) (c : 𝕜)
Goal: ∇ (fun x => c) x = 0
-- gradient_const'
@[simp]
theorem gradient_const' : (∇ fun _ : 𝕜 => c) = fun _ => 0 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] (c : 𝕜)
Goal: (∇ fun x => c) = fun x => 0
-- HasGradientAtFilter.tendsto_nhds
nonrec theorem HasGradientAtFilter.tendsto_nhds (hL : L ≤ 𝓝 x) (h : HasGradientAtFilter f f' x L) :
    Tendsto f L (𝓝 (f x)) 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {L : Filter F} (hL : L ≤ 𝓝 x) (h : HasGradientAtFilter f f' x L)
Goal: Filter.Tendsto f L (𝓝 (f x))
-- HasGradientWithinAt.continuousWithinAt
theorem HasGradientWithinAt.continuousWithinAt (h : HasGradientWithinAt f f' s x) :
    ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} {s : Set F} (h : HasGradientWithinAt f f' s x)
Goal: ContinuousWithinAt f s x
-- HasGradientAt.continuousAt
theorem HasGradientAt.continuousAt (h : HasGradientAt f f' x) : ContinuousAt f x 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {f' x : F} (h : HasGradientAt f f' x)
Goal: ContinuousAt f x
-- HasGradientAt.continuousOn
protected theorem HasGradientAt.continuousOn {f' : F → F} (h : ∀ x ∈ s, HasGradientAt f (f' x) x) :
    ContinuousOn f s 
Arguments: {𝕜 : Type u_1} {F : Type u_2} [RCLike 𝕜] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] {f : F → 𝕜} {s : Set F} {f' : F → F} (h : ∀ x ∈ s, HasGradientAt f (f' x) x)
Goal: ContinuousOn f s
-- HasStrictFDerivAt.const_smul
@[fun_prop]
theorem HasStrictFDerivAt.const_smul (h : HasStrictFDerivAt f f' x) (c : R) :
    HasStrictFDerivAt (fun x => c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : HasStrictFDerivAt f f' x) (c : R)
Goal: HasStrictFDerivAt (fun x => c • f x) (c • f') x
-- HasFDerivAtFilter.const_smul
theorem HasFDerivAtFilter.const_smul (h : HasFDerivAtFilter f f' x L) (c : R) :
    HasFDerivAtFilter (fun x => c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : HasFDerivAtFilter f f' x L) (c : R)
Goal: HasFDerivAtFilter (fun x => c • f x) (c • f') x L
-- HasFDerivWithinAt.const_smul
@[fun_prop]
nonrec theorem HasFDerivWithinAt.const_smul (h : HasFDerivWithinAt f f' s x) (c : R) :
    HasFDerivWithinAt (fun x => c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : HasFDerivWithinAt f f' s x) (c : R)
Goal: HasFDerivWithinAt (fun x => c • f x) (c • f') s x
-- HasFDerivAt.const_smul
@[fun_prop]
nonrec theorem HasFDerivAt.const_smul (h : HasFDerivAt f f' x) (c : R) :
    HasFDerivAt (fun x => c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : HasFDerivAt f f' x) (c : R)
Goal: HasFDerivAt (fun x => c • f x) (c • f') x
-- DifferentiableWithinAt.const_smul
@[fun_prop]
theorem DifferentiableWithinAt.const_smul (h : DifferentiableWithinAt 𝕜 f s x) (c : R) :
    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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : DifferentiableWithinAt 𝕜 f s x) (c : R)
Goal: DifferentiableWithinAt 𝕜 (fun y => c • f y) s x
-- DifferentiableAt.const_smul
@[fun_prop]
theorem DifferentiableAt.const_smul (h : DifferentiableAt 𝕜 f x) (c : R) :
    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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : DifferentiableAt 𝕜 f x) (c : R)
Goal: DifferentiableAt 𝕜 (fun y => c • f y) x
-- DifferentiableOn.const_smul
@[fun_prop]
theorem DifferentiableOn.const_smul (h : DifferentiableOn 𝕜 f s) (c : R) :
    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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : DifferentiableOn 𝕜 f s) (c : R)
Goal: DifferentiableOn 𝕜 (fun y => c • f y) s
-- Differentiable.const_smul
@[fun_prop]
theorem Differentiable.const_smul (h : Differentiable 𝕜 f) (c : R) :
    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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : Differentiable 𝕜 f) (c : R)
Goal: Differentiable 𝕜 fun y => c • f y
-- fderivWithin_const_smul
theorem fderivWithin_const_smul (hxs : UniqueDiffWithinAt 𝕜 s x)
    (h : DifferentiableWithinAt 𝕜 f s x) (c : R) :
    fderivWithin 𝕜 (fun y => c • f y) s x = c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (hxs : UniqueDiffWithinAt 𝕜 s x) (h : DifferentiableWithinAt 𝕜 f s x) (c : R)
Goal: fderivWithin 𝕜 (fun y => c • f y) s x = c • fderivWithin 𝕜 f s x
-- fderiv_const_smul
theorem fderiv_const_smul (h : DifferentiableAt 𝕜 f x) (c : R) :
    fderiv 𝕜 (fun y => c • f y) x = c • 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} {R : Type u_6} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (h : DifferentiableAt 𝕜 f x) (c : R)
Goal: fderiv 𝕜 (fun y => c • f y) x = c • fderiv 𝕜 f x
-- HasStrictFDerivAt.add
@[fun_prop]
nonrec theorem HasStrictFDerivAt.add (hf : HasStrictFDerivAt f f' x)
    (hg : HasStrictFDerivAt g g' x) : HasStrictFDerivAt (fun y => f y + g y) (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 y => f y + g y) (f' + g') x
-- HasFDerivAtFilter.add
theorem HasFDerivAtFilter.add (hf : HasFDerivAtFilter f f' x L)
    (hg : HasFDerivAtFilter g g' x L) : HasFDerivAtFilter (fun y => f y + g y) (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 y => f y + g y) (f' + g') x L
-- HasFDerivWithinAt.add
@[fun_prop]
nonrec theorem HasFDerivWithinAt.add (hf : HasFDerivWithinAt f f' s x)
    (hg : HasFDerivWithinAt g g' s x) : HasFDerivWithinAt (fun y => f y + g y) (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 y => f y + g y) (f' + g') s x