ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- HasDerivWithinAt.congr_of_eventuallyEq_of_mem
theorem HasDerivWithinAt.congr_of_eventuallyEq_of_mem (h : HasDerivWithinAt f f' s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) : HasDerivWithinAt f₁ f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s)
Goal: HasDerivWithinAt f₁ f' s x
-- Filter.EventuallyEq.hasDerivWithinAt_iff_of_mem
theorem Filter.EventuallyEq.hasDerivWithinAt_iff_of_mem (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) :
    HasDerivWithinAt f₁ f' s x ↔ HasDerivWithinAt f f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s)
Goal: HasDerivWithinAt f₁ f' s x ↔ HasDerivWithinAt f f' s x
-- HasStrictDerivAt.congr_deriv
theorem HasStrictDerivAt.congr_deriv (h : HasStrictDerivAt f f' x) (h' : f' = g') :
    HasStrictDerivAt f g' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' g' : F} {x : 𝕜} (h : HasStrictDerivAt f f' x) (h' : f' = g')
Goal: HasStrictDerivAt f g' x
-- HasDerivAt.congr_deriv
theorem HasDerivAt.congr_deriv (h : HasDerivAt f f' x) (h' : f' = g') : HasDerivAt f g' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' g' : F} {x : 𝕜} (h : HasDerivAt f f' x) (h' : f' = g')
Goal: HasDerivAt f g' x
-- HasDerivWithinAt.congr_deriv
theorem HasDerivWithinAt.congr_deriv (h : HasDerivWithinAt f f' s x) (h' : f' = g') :
    HasDerivWithinAt f g' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' g' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x) (h' : f' = g')
Goal: HasDerivWithinAt f g' s x
-- HasDerivAt.congr_of_eventuallyEq
theorem HasDerivAt.congr_of_eventuallyEq (h : HasDerivAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasDerivAt f₁ f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f)
Goal: HasDerivAt f₁ f' x
-- Filter.EventuallyEq.hasDerivAt_iff
theorem Filter.EventuallyEq.hasDerivAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
    HasDerivAt f₀ f' x ↔ HasDerivAt f₁ f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f₀ f₁ : 𝕜 → F} {f' : F} {x : 𝕜} (h : f₀ =ᶠ[𝓝 x] f₁)
Goal: HasDerivAt f₀ f' x ↔ HasDerivAt f₁ f' x
-- Filter.EventuallyEq.derivWithin_eq
theorem Filter.EventuallyEq.derivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    derivWithin f₁ s x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: derivWithin f₁ s x = derivWithin f s x
-- derivWithin_congr
theorem derivWithin_congr (hs : EqOn f₁ f s) (hx : f₁ x = f x) :
    derivWithin f₁ s x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hs : Set.EqOn f₁ f s) (hx : f₁ x = f x)
Goal: derivWithin f₁ s x = derivWithin f s x
-- Filter.EventuallyEq.deriv_eq
theorem Filter.EventuallyEq.deriv_eq (hL : f₁ =ᶠ[𝓝 x] f) : deriv f₁ x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {x : 𝕜} (hL : f₁ =ᶠ[𝓝 x] f)
Goal: deriv f₁ x = deriv f x
-- Filter.EventuallyEq.deriv
protected theorem Filter.EventuallyEq.deriv (h : f₁ =ᶠ[𝓝 x] f) : deriv f₁ =ᶠ[𝓝 x] deriv f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : 𝕜 → F} {x : 𝕜} (h : f₁ =ᶠ[𝓝 x] f)
Goal: deriv f₁ =ᶠ[𝓝 x] deriv f
-- hasDerivAtFilter_id
theorem hasDerivAtFilter_id : HasDerivAtFilter id 1 x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (L : Filter 𝕜)
Goal: HasDerivAtFilter id 1 x L
-- hasDerivWithinAt_id
theorem hasDerivWithinAt_id : HasDerivWithinAt id 1 s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (s : Set 𝕜)
Goal: HasDerivWithinAt id 1 s x
-- hasDerivAt_id
theorem hasDerivAt_id : HasDerivAt id 1 x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasDerivAt id 1 x
-- hasDerivAt_id'
theorem hasDerivAt_id' : HasDerivAt (fun x : 𝕜 => x) 1 x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasDerivAt (fun x => x) 1 x
-- hasStrictDerivAt_id
theorem hasStrictDerivAt_id : HasStrictDerivAt id 1 x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasStrictDerivAt id 1 x
-- deriv_id
theorem deriv_id : deriv id x = 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: deriv id x = 1
-- deriv_id'
@[simp]
theorem deriv_id' : deriv (@id 𝕜) = fun _ => 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: deriv id = fun x => 1
-- deriv_id''
@[simp]
theorem deriv_id'' : (deriv fun x : 𝕜 => x) = fun _ => 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜]
Goal: (deriv fun x => x) = fun x => 1
-- derivWithin_id
theorem derivWithin_id (hxs : UniqueDiffWithinAt 𝕜 s x) : derivWithin id s x = 1 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (s : Set 𝕜) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin id s x = 1
-- hasDerivAtFilter_const
theorem hasDerivAtFilter_const : HasDerivAtFilter (fun _ => c) 0 x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (L : Filter 𝕜) (c : F)
Goal: HasDerivAtFilter (fun x => c) 0 x L
-- hasStrictDerivAt_const
theorem hasStrictDerivAt_const : HasStrictDerivAt (fun _ => c) 0 x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (c : F)
Goal: HasStrictDerivAt (fun x => c) 0 x
-- hasDerivWithinAt_const
theorem hasDerivWithinAt_const : HasDerivWithinAt (fun _ => c) 0 s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (s : Set 𝕜) (c : F)
Goal: HasDerivWithinAt (fun x => c) 0 s x
-- hasDerivAt_const
theorem hasDerivAt_const : HasDerivAt (fun _ => c) 0 x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (c : F)
Goal: HasDerivAt (fun x => c) 0 x
-- deriv_const
theorem deriv_const : deriv (fun _ => c) x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (c : F)
Goal: deriv (fun x => c) x = 0
-- deriv_const'
@[simp]
theorem deriv_const' : (deriv fun _ : 𝕜 => c) = fun _ => 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (c : F)
Goal: (deriv fun x => c) = fun x => 0
-- derivWithin_const
theorem derivWithin_const (hxs : UniqueDiffWithinAt 𝕜 s x) : derivWithin (fun _ => c) s x = 0 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) (s : Set 𝕜) (c : F) (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun x => c) s x = 0
-- HasDerivAtFilter.tendsto_nhds
nonrec theorem HasDerivAtFilter.tendsto_nhds (hL : L ≤ 𝓝 x) (h : HasDerivAtFilter f f' x L) :
    Tendsto f L (𝓝 (f x)) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (hL : L ≤ 𝓝 x) (h : HasDerivAtFilter f f' x L)
Goal: Filter.Tendsto f L (𝓝 (f x))
-- HasDerivWithinAt.continuousWithinAt
theorem HasDerivWithinAt.continuousWithinAt (h : HasDerivWithinAt f f' s x) :
    ContinuousWithinAt f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x)
Goal: ContinuousWithinAt f s x
-- HasDerivAt.continuousAt
theorem HasDerivAt.continuousAt (h : HasDerivAt f f' x) : ContinuousAt f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x)
Goal: ContinuousAt f x
-- HasDerivAt.continuousOn
protected theorem HasDerivAt.continuousOn {f f' : 𝕜 → F} (hderiv : ∀ x ∈ s, HasDerivAt f (f' x) x) :
    ContinuousOn f s 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s : Set 𝕜} {f f' : 𝕜 → F} (hderiv : ∀ x ∈ s, HasDerivAt f (f' x) x)
Goal: ContinuousOn f s
-- HasDerivAt.le_of_lip'
theorem HasDerivAt.le_of_lip' {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
    {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
    ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀) {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ (x : 𝕜) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖)
Goal: ‖f'‖ ≤ C
-- HasDerivAt.le_of_lipschitzOn
theorem HasDerivAt.le_of_lipschitzOn {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
    {s : Set 𝕜} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀) {s : Set 𝕜} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖f'‖ ≤ ↑C
-- HasDerivAt.le_of_lipschitz
theorem HasDerivAt.le_of_lipschitz {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀)
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖f'‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x₀ : 𝕜} (hf : HasDerivAt f f' x₀) {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖f'‖ ≤ ↑C
-- norm_deriv_le_of_lip'
theorem norm_deriv_le_of_lip' {f : 𝕜 → F} {x₀ : 𝕜}
    {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
    ‖deriv f x₀‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x₀ : 𝕜} {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ (x : 𝕜) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖)
Goal: ‖deriv f x₀‖ ≤ C
-- norm_deriv_le_of_lipschitzOn
theorem norm_deriv_le_of_lipschitzOn {f : 𝕜 → F} {x₀ : 𝕜} {s : Set 𝕜} (hs : s ∈ 𝓝 x₀)
    {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖deriv f x₀‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x₀ : 𝕜} {s : Set 𝕜} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖deriv f x₀‖ ≤ ↑C
-- norm_deriv_le_of_lipschitz
theorem norm_deriv_le_of_lipschitz {f : 𝕜 → F} {x₀ : 𝕜}
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖deriv f x₀‖ ≤ C 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x₀ : 𝕜} {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖deriv f x₀‖ ≤ ↑C
-- HasDerivAtFilter.add
nonrec theorem HasDerivAtFilter.add (hf : HasDerivAtFilter f f' x L)
    (hg : HasDerivAtFilter g g' x L) : HasDerivAtFilter (fun y => f y + g y) (f' + g') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} {L : Filter 𝕜} (hf : HasDerivAtFilter f f' x L) (hg : HasDerivAtFilter g g' x L)
Goal: HasDerivAtFilter (fun y => f y + g y) (f' + g') x L
-- HasStrictDerivAt.add
nonrec theorem HasStrictDerivAt.add (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x) :
    HasStrictDerivAt (fun y => f y + g y) (f' + g') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} (hf : HasStrictDerivAt f f' x) (hg : HasStrictDerivAt g g' x)
Goal: HasStrictDerivAt (fun y => f y + g y) (f' + g') x
-- HasDerivWithinAt.add
nonrec theorem HasDerivWithinAt.add (hf : HasDerivWithinAt f f' s x)
    (hg : HasDerivWithinAt g g' s x) : HasDerivWithinAt (fun y => f y + g y) (f' + g') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} {s : Set 𝕜} (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
Goal: HasDerivWithinAt (fun y => f y + g y) (f' + g') s x
-- HasDerivAt.add
nonrec theorem HasDerivAt.add (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) :
    HasDerivAt (fun x => f x + g x) (f' + g') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {f' g' : F} {x : 𝕜} (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x)
Goal: HasDerivAt (fun x => f x + g x) (f' + g') x
-- derivWithin_add
theorem derivWithin_add (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x)
    (hg : DifferentiableWithinAt 𝕜 g s x) :
    derivWithin (fun y => f y + g y) s x = derivWithin f s x + derivWithin g s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (hf : DifferentiableWithinAt 𝕜 f s x) (hg : DifferentiableWithinAt 𝕜 g s x)
Goal: derivWithin (fun y => f y + g y) s x = derivWithin f s x + derivWithin g s x
-- deriv_add
@[simp]
theorem deriv_add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) :
    deriv (fun y => f y + g y) x = deriv f x + deriv g x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : 𝕜 → F} {x : 𝕜} (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x)
Goal: deriv (fun y => f y + g y) x = deriv f x + deriv g x
-- HasStrictDerivAt.add_const
theorem HasStrictDerivAt.add_const (c : F) (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun y ↦ f y + c) f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (c : F) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun y => f y + c) f' x
-- HasDerivAtFilter.add_const
theorem HasDerivAtFilter.add_const (hf : HasDerivAtFilter f f' x L) (c : F) :
    HasDerivAtFilter (fun y => f y + c) f' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (hf : HasDerivAtFilter f f' x L) (c : F)
Goal: HasDerivAtFilter (fun y => f y + c) f' x L
-- HasDerivWithinAt.add_const
nonrec theorem HasDerivWithinAt.add_const (hf : HasDerivWithinAt f f' s x) (c : F) :
    HasDerivWithinAt (fun y => f y + c) f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (hf : HasDerivWithinAt f f' s x) (c : F)
Goal: HasDerivWithinAt (fun y => f y + c) f' s x
-- HasDerivAt.add_const
nonrec theorem HasDerivAt.add_const (hf : HasDerivAt f f' x) (c : F) :
    HasDerivAt (fun x => f x + c) f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (hf : HasDerivAt f f' x) (c : F)
Goal: HasDerivAt (fun x => f x + c) f' x
-- derivWithin_add_const
theorem derivWithin_add_const (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    derivWithin (fun y => f y + c) s x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: derivWithin (fun y => f y + c) s x = derivWithin f s x
-- deriv_add_const
theorem deriv_add_const (c : F) : deriv (fun y => f y + c) x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (c : F)
Goal: deriv (fun y => f y + c) x = deriv f x
-- deriv_add_const'
@[simp]
theorem deriv_add_const' (c : F) : (deriv fun y => f y + c) = deriv f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} (c : F)
Goal: (deriv fun y => f y + c) = deriv f
-- HasStrictDerivAt.const_add
theorem HasStrictDerivAt.const_add (c : F) (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun y ↦ c + f y) f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (c : F) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun y => c + f y) f' x
-- HasDerivAtFilter.const_add
theorem HasDerivAtFilter.const_add (c : F) (hf : HasDerivAtFilter f f' x L) :
    HasDerivAtFilter (fun y => c + f y) f' x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (c : F) (hf : HasDerivAtFilter f f' x L)
Goal: HasDerivAtFilter (fun y => c + f y) f' x L
-- HasDerivWithinAt.const_add
nonrec theorem HasDerivWithinAt.const_add (c : F) (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun y => c + f y) f' s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (c : F) (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun y => c + f y) f' s x
-- HasDerivAt.const_add
nonrec theorem HasDerivAt.const_add (c : F) (hf : HasDerivAt f f' x) :
    HasDerivAt (fun x => c + f x) f' x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (c : F) (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => c + f x) f' x
-- derivWithin_const_add
theorem derivWithin_const_add (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
    derivWithin (fun y => c + f y) s x = derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F)
Goal: derivWithin (fun y => c + f y) s x = derivWithin f s x
-- deriv_const_add
theorem deriv_const_add (c : F) : deriv (fun y => c + f y) x = deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} (c : F)
Goal: deriv (fun y => c + f y) x = deriv f x
-- deriv_const_add'
@[simp]
theorem deriv_const_add' (c : F) : (deriv fun y => c + f y) = deriv f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} (c : F)
Goal: (deriv fun y => c + f y) = deriv f
-- HasDerivAtFilter.sum
theorem HasDerivAtFilter.sum (h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L) :
    HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {L : Filter 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} {A' : ι → F} (h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L)
Goal: HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L
-- HasStrictDerivAt.sum
theorem HasStrictDerivAt.sum (h : ∀ i ∈ u, HasStrictDerivAt (A i) (A' i) x) :
    HasStrictDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} {A' : ι → F} (h : ∀ i ∈ u, HasStrictDerivAt (A i) (A' i) x)
Goal: HasStrictDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x
-- HasDerivWithinAt.sum
theorem HasDerivWithinAt.sum (h : ∀ i ∈ u, HasDerivWithinAt (A i) (A' i) s x) :
    HasDerivWithinAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} {A' : ι → F} (h : ∀ i ∈ u, HasDerivWithinAt (A i) (A' i) s x)
Goal: HasDerivWithinAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) s x
-- HasDerivAt.sum
theorem HasDerivAt.sum (h : ∀ i ∈ u, HasDerivAt (A i) (A' i) x) :
    HasDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} {A' : ι → F} (h : ∀ i ∈ u, HasDerivAt (A i) (A' i) x)
Goal: HasDerivAt (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x
-- derivWithin_sum
theorem derivWithin_sum (hxs : UniqueDiffWithinAt 𝕜 s x)
    (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) :
    derivWithin (fun y => ∑ i ∈ u, A i y) s x = ∑ i ∈ u, derivWithin (A i) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} (hxs : UniqueDiffWithinAt 𝕜 s x) (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x)
Goal: derivWithin (fun y => ∑ i ∈ u, A i y) s x = ∑ i ∈ u, derivWithin (A i) s x
-- deriv_sum
@[simp]
theorem deriv_sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) :
    deriv (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, deriv (A i) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {ι : Type u_1} {u : Finset ι} {A : ι → 𝕜 → F} (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x)
Goal: deriv (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, deriv (A i) x
-- HasDerivAtFilter.neg
nonrec theorem HasDerivAtFilter.neg (h : HasDerivAtFilter f f' x L) :
    HasDerivAtFilter (fun x => -f x) (-f') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} (h : HasDerivAtFilter f f' x L)
Goal: HasDerivAtFilter (fun x => -f x) (-f') x L
-- HasDerivWithinAt.neg
nonrec theorem HasDerivWithinAt.neg (h : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun x => -f x) (-f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} (h : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun x => -f x) (-f') s x
-- HasDerivAt.neg
nonrec theorem HasDerivAt.neg (h : HasDerivAt f f' x) : HasDerivAt (fun x => -f x) (-f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasDerivAt f f' x)
Goal: HasDerivAt (fun x => -f x) (-f') x
-- HasStrictDerivAt.neg
nonrec theorem HasStrictDerivAt.neg (h : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun x => -f x) (-f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} (h : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun x => -f x) (-f') x
-- derivWithin.neg
theorem derivWithin.neg (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (fun y => -f y) s x = -derivWithin f s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (fun y => -f y) s x = -derivWithin f s x
-- deriv.neg
theorem deriv.neg : deriv (fun y => -f y) x = -deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜}
Goal: deriv (fun y => -f y) x = -deriv f x
-- deriv.neg'
@[simp]
theorem deriv.neg' : (deriv fun y => -f y) = fun x => -deriv f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F}
Goal: (deriv fun y => -f y) = fun x => -deriv f x
-- hasDerivAtFilter_neg
theorem hasDerivAtFilter_neg : HasDerivAtFilter Neg.neg (-1) x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (L : Filter 𝕜)
Goal: HasDerivAtFilter Neg.neg (-1) x L
-- hasDerivWithinAt_neg
theorem hasDerivWithinAt_neg : HasDerivWithinAt Neg.neg (-1) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) (s : Set 𝕜)
Goal: HasDerivWithinAt Neg.neg (-1) s x
-- hasDerivAt_neg
theorem hasDerivAt_neg : HasDerivAt Neg.neg (-1) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜)
Goal: HasDerivAt Neg.neg (-1) x