ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Differentiable.continuous
@[fun_prop]
theorem Differentiable.continuous (h : Differentiable 𝕜 f) : Continuous 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} (h : Differentiable 𝕜 f)
Goal: Continuous f
-- HasStrictFDerivAt.continuousAt
protected theorem HasStrictFDerivAt.continuousAt (hf : HasStrictFDerivAt 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} (hf : HasStrictFDerivAt f f' x)
Goal: ContinuousAt f x
-- HasStrictFDerivAt.isBigO_sub_rev
theorem HasStrictFDerivAt.isBigO_sub_rev {f' : E ≃L[𝕜] F}
    (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) x) :
    (fun p : E × E => p.1 - p.2) =O[𝓝 (x, x)] fun p : E × E => f p.1 - f p.2 
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} {f' : E ≃L[𝕜] F} (hf : HasStrictFDerivAt f (↑f') x)
Goal: (fun p => p.1 - p.2) =O[𝓝 (x, x)] fun p => f p.1 - f p.2
-- HasFDerivAtFilter.isBigO_sub_rev
theorem HasFDerivAtFilter.isBigO_sub_rev (hf : HasFDerivAtFilter f f' x L) {C}
    (hf' : AntilipschitzWith C f') : (fun x' => x' - x) =O[L] fun x' => f x' - f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (hf : HasFDerivAtFilter f f' x L) {C : ℝ≥0} (hf' : AntilipschitzWith C ⇑f')
Goal: (fun x' => x' - x) =O[L] fun x' => f x' - f x
-- hasFDerivWithinAt_congr_set'
theorem hasFDerivWithinAt_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f 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} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' t x
-- hasFDerivWithinAt_congr_set
theorem hasFDerivWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
    HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f 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} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : s =ᶠ[𝓝 x] t)
Goal: HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' t x
-- differentiableWithinAt_congr_set'
theorem differentiableWithinAt_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    DifferentiableWithinAt 𝕜 f s 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 t : Set E} (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: DifferentiableWithinAt 𝕜 f s x ↔ DifferentiableWithinAt 𝕜 f t x
-- differentiableWithinAt_congr_set
theorem differentiableWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
    DifferentiableWithinAt 𝕜 f s 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 t : Set E} (h : s =ᶠ[𝓝 x] t)
Goal: DifferentiableWithinAt 𝕜 f s x ↔ DifferentiableWithinAt 𝕜 f t x
-- fderivWithin_congr_set'
theorem fderivWithin_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    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} (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x
-- fderivWithin_congr_set
theorem fderivWithin_congr_set (h : s =ᶠ[𝓝 x] t) : 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} (h : s =ᶠ[𝓝 x] t)
Goal: fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x
-- fderivWithin_eventually_congr_set'
theorem fderivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
    fderivWithin 𝕜 f s =ᶠ[𝓝 x] fderivWithin 𝕜 f t 
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} (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t)
Goal: fderivWithin 𝕜 f s =ᶠ[𝓝 x] fderivWithin 𝕜 f t
-- fderivWithin_eventually_congr_set
theorem fderivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) :
    fderivWithin 𝕜 f s =ᶠ[𝓝 x] fderivWithin 𝕜 f t 
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 : s =ᶠ[𝓝 x] t)
Goal: fderivWithin 𝕜 f s =ᶠ[𝓝 x] fderivWithin 𝕜 f t
-- Filter.EventuallyEq.hasStrictFDerivAt_iff
theorem Filter.EventuallyEq.hasStrictFDerivAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) (h' : ∀ y, f₀' y = f₁' y) :
    HasStrictFDerivAt f₀ f₀' x ↔ HasStrictFDerivAt f₁ f₁' 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] {f₀ f₁ : E → F} {f₀' f₁' : E →L[𝕜] F} {x : E} (h : f₀ =ᶠ[𝓝 x] f₁) (h' : ∀ (y : E), f₀' y = f₁' y)
Goal: HasStrictFDerivAt f₀ f₀' x ↔ HasStrictFDerivAt f₁ f₁' x
-- HasStrictFDerivAt.congr_fderiv
theorem HasStrictFDerivAt.congr_fderiv (h : HasStrictFDerivAt f f' x) (h' : f' = g') :
    HasStrictFDerivAt 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 : E → F} {f' g' : E →L[𝕜] F} {x : E} (h : HasStrictFDerivAt f f' x) (h' : f' = g')
Goal: HasStrictFDerivAt f g' x
-- HasFDerivAt.congr_fderiv
theorem HasFDerivAt.congr_fderiv (h : HasFDerivAt f f' x) (h' : f' = g') : HasFDerivAt 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 : E → F} {f' g' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x) (h' : f' = g')
Goal: HasFDerivAt f g' x
-- HasFDerivWithinAt.congr_fderiv
theorem HasFDerivWithinAt.congr_fderiv (h : HasFDerivWithinAt f f' s x) (h' : f' = g') :
    HasFDerivWithinAt 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 : E → F} {f' g' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (h' : f' = g')
Goal: HasFDerivWithinAt f g' s x
-- HasStrictFDerivAt.congr_of_eventuallyEq
theorem HasStrictFDerivAt.congr_of_eventuallyEq (h : HasStrictFDerivAt f f' x)
    (h₁ : f =ᶠ[𝓝 x] f₁) : HasStrictFDerivAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasStrictFDerivAt f f' x) (h₁ : f =ᶠ[𝓝 x] f₁)
Goal: HasStrictFDerivAt f₁ f' x
-- Filter.EventuallyEq.hasFDerivAtFilter_iff
theorem Filter.EventuallyEq.hasFDerivAtFilter_iff (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x)
    (h₁ : ∀ x, f₀' x = f₁' x) : HasFDerivAtFilter f₀ f₀' x L ↔ HasFDerivAtFilter f₁ f₁' 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] {f₀ f₁ : E → F} {f₀' f₁' : E →L[𝕜] F} {x : E} {L : Filter E} (h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : ∀ (x : E), f₀' x = f₁' x)
Goal: HasFDerivAtFilter f₀ f₀' x L ↔ HasFDerivAtFilter f₁ f₁' x L
-- HasFDerivAtFilter.congr_of_eventuallyEq
theorem HasFDerivAtFilter.congr_of_eventuallyEq (h : HasFDerivAtFilter f f' x L) (hL : f₁ =ᶠ[L] f)
    (hx : f₁ x = f x) : HasFDerivAtFilter f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (h : HasFDerivAtFilter f f' x L) (hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x)
Goal: HasFDerivAtFilter f₁ f' x L
-- Filter.EventuallyEq.hasFDerivAt_iff
theorem Filter.EventuallyEq.hasFDerivAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
    HasFDerivAt f₀ f' x ↔ HasFDerivAt f₁ 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] {f₀ f₁ : E → F} {f' : E →L[𝕜] F} {x : E} (h : f₀ =ᶠ[𝓝 x] f₁)
Goal: HasFDerivAt f₀ f' x ↔ HasFDerivAt f₁ f' x
-- Filter.EventuallyEq.differentiableAt_iff
theorem Filter.EventuallyEq.differentiableAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
    DifferentiableAt 𝕜 f₀ x ↔ DifferentiableAt 𝕜 f₁ 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] {f₀ f₁ : E → F} {x : E} (h : f₀ =ᶠ[𝓝 x] f₁)
Goal: DifferentiableAt 𝕜 f₀ x ↔ DifferentiableAt 𝕜 f₁ x
-- Filter.EventuallyEq.hasFDerivWithinAt_iff
theorem Filter.EventuallyEq.hasFDerivWithinAt_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
    HasFDerivWithinAt f₀ f' s x ↔ HasFDerivWithinAt f₁ 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] {f₀ f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x)
Goal: HasFDerivWithinAt f₀ f' s x ↔ HasFDerivWithinAt f₁ f' s x
-- Filter.EventuallyEq.hasFDerivWithinAt_iff_of_mem
theorem Filter.EventuallyEq.hasFDerivWithinAt_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
    HasFDerivWithinAt f₀ f' s x ↔ HasFDerivWithinAt f₁ 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] {f₀ f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s)
Goal: HasFDerivWithinAt f₀ f' s x ↔ HasFDerivWithinAt f₁ f' s x
-- Filter.EventuallyEq.differentiableWithinAt_iff
theorem Filter.EventuallyEq.differentiableWithinAt_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
    DifferentiableWithinAt 𝕜 f₀ s x ↔ DifferentiableWithinAt 𝕜 f₁ 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] {f₀ f₁ : E → F} {x : E} {s : Set E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x)
Goal: DifferentiableWithinAt 𝕜 f₀ s x ↔ DifferentiableWithinAt 𝕜 f₁ s x
-- Filter.EventuallyEq.differentiableWithinAt_iff_of_mem
theorem Filter.EventuallyEq.differentiableWithinAt_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
    DifferentiableWithinAt 𝕜 f₀ s x ↔ DifferentiableWithinAt 𝕜 f₁ 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] {f₀ f₁ : E → F} {x : E} {s : Set E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s)
Goal: DifferentiableWithinAt 𝕜 f₀ s x ↔ DifferentiableWithinAt 𝕜 f₁ s x
-- HasFDerivWithinAt.congr_mono
theorem HasFDerivWithinAt.congr_mono (h : HasFDerivWithinAt f f' s x) (ht : EqOn f₁ f t)
    (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasFDerivWithinAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' s x) (ht : Set.EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: HasFDerivWithinAt f₁ f' t x
-- HasFDerivWithinAt.congr
theorem HasFDerivWithinAt.congr (h : HasFDerivWithinAt f f' s x) (hs : EqOn f₁ f s)
    (hx : f₁ x = f x) : HasFDerivWithinAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hs : Set.EqOn f₁ f s) (hx : f₁ x = f x)
Goal: HasFDerivWithinAt f₁ f' s x
-- HasFDerivWithinAt.congr'
theorem HasFDerivWithinAt.congr' (h : HasFDerivWithinAt f f' s x) (hs : EqOn f₁ f s) (hx : x ∈ s) :
    HasFDerivWithinAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (hs : Set.EqOn f₁ f s) (hx : x ∈ s)
Goal: HasFDerivWithinAt f₁ f' s x
-- HasFDerivWithinAt.congr_of_eventuallyEq
theorem HasFDerivWithinAt.congr_of_eventuallyEq (h : HasFDerivWithinAt f f' s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasFDerivWithinAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasFDerivWithinAt f₁ f' s x
-- HasFDerivAt.congr_of_eventuallyEq
theorem HasFDerivAt.congr_of_eventuallyEq (h : HasFDerivAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasFDerivAt f₁ 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 f₁ : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x) (h₁ : f₁ =ᶠ[𝓝 x] f)
Goal: HasFDerivAt f₁ f' x
-- DifferentiableWithinAt.congr_mono
theorem DifferentiableWithinAt.congr_mono (h : DifferentiableWithinAt 𝕜 f s x) (ht : EqOn f₁ f t)
    (hx : f₁ x = f x) (h₁ : t ⊆ s) : DifferentiableWithinAt 𝕜 f₁ 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 f₁ : E → F} {x : E} {s t : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (ht : Set.EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: DifferentiableWithinAt 𝕜 f₁ t x
-- DifferentiableWithinAt.congr
theorem DifferentiableWithinAt.congr (h : DifferentiableWithinAt 𝕜 f s x) (ht : ∀ x ∈ s, f₁ x = f x)
    (hx : f₁ x = f x) : DifferentiableWithinAt 𝕜 f₁ 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 f₁ : E → F} {x : E} {s : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (ht : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: DifferentiableWithinAt 𝕜 f₁ s x
-- DifferentiableWithinAt.congr_of_eventuallyEq
theorem DifferentiableWithinAt.congr_of_eventuallyEq (h : DifferentiableWithinAt 𝕜 f s x)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : DifferentiableWithinAt 𝕜 f₁ 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 f₁ : E → F} {x : E} {s : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: DifferentiableWithinAt 𝕜 f₁ s x
-- DifferentiableOn.congr_mono
theorem DifferentiableOn.congr_mono (h : DifferentiableOn 𝕜 f s) (h' : ∀ x ∈ t, f₁ x = f x)
    (h₁ : t ⊆ s) : DifferentiableOn 𝕜 f₁ t 
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 f₁ : E → F} {s t : Set E} (h : DifferentiableOn 𝕜 f s) (h' : ∀ x ∈ t, f₁ x = f x) (h₁ : t ⊆ s)
Goal: DifferentiableOn 𝕜 f₁ t
-- DifferentiableOn.congr
theorem DifferentiableOn.congr (h : DifferentiableOn 𝕜 f s) (h' : ∀ x ∈ s, f₁ x = f x) :
    DifferentiableOn 𝕜 f₁ 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 f₁ : E → F} {s : Set E} (h : DifferentiableOn 𝕜 f s) (h' : ∀ x ∈ s, f₁ x = f x)
Goal: DifferentiableOn 𝕜 f₁ s
-- differentiableOn_congr
theorem differentiableOn_congr (h' : ∀ x ∈ s, f₁ x = f x) :
    DifferentiableOn 𝕜 f₁ s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {s : Set E} (h' : ∀ x ∈ s, f₁ x = f x)
Goal: DifferentiableOn 𝕜 f₁ s ↔ DifferentiableOn 𝕜 f s
-- DifferentiableAt.congr_of_eventuallyEq
theorem DifferentiableAt.congr_of_eventuallyEq (h : DifferentiableAt 𝕜 f x) (hL : f₁ =ᶠ[𝓝 x] f) :
    DifferentiableAt 𝕜 f₁ 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 f₁ : E → F} {x : E} (h : DifferentiableAt 𝕜 f x) (hL : f₁ =ᶠ[𝓝 x] f)
Goal: DifferentiableAt 𝕜 f₁ x
-- DifferentiableWithinAt.fderivWithin_congr_mono
theorem DifferentiableWithinAt.fderivWithin_congr_mono (h : DifferentiableWithinAt 𝕜 f s x)
    (hs : EqOn f₁ f t) (hx : f₁ x = f x) (hxt : UniqueDiffWithinAt 𝕜 t x) (h₁ : t ⊆ s) :
    fderivWithin 𝕜 f₁ 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 f₁ : E → F} {x : E} {s t : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (hs : Set.EqOn f₁ f t) (hx : f₁ x = f x) (hxt : UniqueDiffWithinAt 𝕜 t x) (h₁ : t ⊆ s)
Goal: fderivWithin 𝕜 f₁ t x = fderivWithin 𝕜 f s x
-- Filter.EventuallyEq.fderivWithin_eq
theorem Filter.EventuallyEq.fderivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {s : Set E} (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x
-- Filter.EventuallyEq.fderivWithin'
theorem Filter.EventuallyEq.fderivWithin' (hs : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) :
    fderivWithin 𝕜 f₁ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 f t 
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 f₁ : E → F} {x : E} {s t : Set E} (hs : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s)
Goal: fderivWithin 𝕜 f₁ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 f t
-- Filter.EventuallyEq.fderivWithin
protected theorem Filter.EventuallyEq.fderivWithin (hs : f₁ =ᶠ[𝓝[s] x] f) :
    fderivWithin 𝕜 f₁ s =ᶠ[𝓝[s] x] fderivWithin 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 f₁ : E → F} {x : E} {s : Set E} (hs : f₁ =ᶠ[𝓝[s] x] f)
Goal: fderivWithin 𝕜 f₁ s =ᶠ[𝓝[s] x] fderivWithin 𝕜 f s
-- Filter.EventuallyEq.fderivWithin_eq_nhds
theorem Filter.EventuallyEq.fderivWithin_eq_nhds (h : f₁ =ᶠ[𝓝 x] f) :
    fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {s : Set E} (h : f₁ =ᶠ[𝓝 x] f)
Goal: fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x
-- fderivWithin_congr
theorem fderivWithin_congr (hs : EqOn f₁ f s) (hx : f₁ x = f x) :
    fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {s : Set E} (hs : Set.EqOn f₁ f s) (hx : f₁ x = f x)
Goal: fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x
-- fderivWithin_congr'
theorem fderivWithin_congr' (hs : EqOn f₁ f s) (hx : x ∈ s) :
    fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {s : Set E} (hs : Set.EqOn f₁ f s) (hx : x ∈ s)
Goal: fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x
-- Filter.EventuallyEq.fderiv_eq
theorem Filter.EventuallyEq.fderiv_eq (h : f₁ =ᶠ[𝓝 x] f) : fderiv 𝕜 f₁ 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 f₁ : E → F} {x : E} (h : f₁ =ᶠ[𝓝 x] f)
Goal: fderiv 𝕜 f₁ x = fderiv 𝕜 f x
-- Filter.EventuallyEq.fderiv
protected theorem Filter.EventuallyEq.fderiv (h : f₁ =ᶠ[𝓝 x] f) : fderiv 𝕜 f₁ =ᶠ[𝓝 x] 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 f₁ : E → F} {x : E} (h : f₁ =ᶠ[𝓝 x] f)
Goal: fderiv 𝕜 f₁ =ᶠ[𝓝 x] fderiv 𝕜 f
-- hasStrictFDerivAt_id
@[fun_prop]
theorem hasStrictFDerivAt_id (x : E) : HasStrictFDerivAt id (id 𝕜 E) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: HasStrictFDerivAt id (ContinuousLinearMap.id 𝕜 E) x
-- hasFDerivAtFilter_id
theorem hasFDerivAtFilter_id (x : E) (L : Filter E) : HasFDerivAtFilter id (id 𝕜 E) x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (L : Filter E)
Goal: HasFDerivAtFilter id (ContinuousLinearMap.id 𝕜 E) x L
-- hasFDerivWithinAt_id
@[fun_prop]
theorem hasFDerivWithinAt_id (x : E) (s : Set E) : HasFDerivWithinAt id (id 𝕜 E) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E) (s : Set E)
Goal: HasFDerivWithinAt id (ContinuousLinearMap.id 𝕜 E) s x
-- hasFDerivAt_id
@[fun_prop]
theorem hasFDerivAt_id (x : E) : HasFDerivAt id (id 𝕜 E) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: HasFDerivAt id (ContinuousLinearMap.id 𝕜 E) x
-- differentiableAt_id
@[simp, fun_prop]
theorem differentiableAt_id : DifferentiableAt 𝕜 id x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: DifferentiableAt 𝕜 id x
-- differentiableAt_id'
@[simp]
theorem differentiableAt_id' : DifferentiableAt 𝕜 (fun x => x) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: DifferentiableAt 𝕜 (fun x => x) x
-- differentiableWithinAt_id
@[fun_prop]
theorem differentiableWithinAt_id : DifferentiableWithinAt 𝕜 id s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E} {s : Set E}
Goal: DifferentiableWithinAt 𝕜 id s x
-- differentiable_id
@[simp, fun_prop]
theorem differentiable_id : Differentiable 𝕜 (id : E → E) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: Differentiable 𝕜 id
-- differentiable_id'
@[simp]
theorem differentiable_id' : Differentiable 𝕜 fun x : E => x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
Goal: Differentiable 𝕜 fun x => x
-- differentiableOn_id
@[fun_prop]
theorem differentiableOn_id : DifferentiableOn 𝕜 id s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {s : Set E}
Goal: DifferentiableOn 𝕜 id s
-- fderiv_id
@[simp]
theorem fderiv_id : fderiv 𝕜 id x = id 𝕜 E 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {x : E}
Goal: fderiv 𝕜 id x = ContinuousLinearMap.id 𝕜 E