ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LineDifferentiableWithinAt.lineDifferentiableAt
theorem LineDifferentiableWithinAt.lineDifferentiableAt (h : LineDifferentiableWithinAt 𝕜 f s x v)
    (hs : s ∈ 𝓝 x) : LineDifferentiableAt 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (hs : s ∈ 𝓝 x)
Goal: LineDifferentiableAt 𝕜 f x v
-- HasFDerivWithinAt.hasLineDerivWithinAt
lemma HasFDerivWithinAt.hasLineDerivWithinAt (hf : HasFDerivWithinAt f L s x) (v : E) :
    HasLineDerivWithinAt 𝕜 f (L v) s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s : Set E} {x : E} {L : E →L[𝕜] F} (hf : HasFDerivWithinAt f L s x) (v : E)
Goal: HasLineDerivWithinAt 𝕜 f (L v) s x v
-- HasFDerivAt.hasLineDerivAt
lemma HasFDerivAt.hasLineDerivAt (hf : HasFDerivAt f L x) (v : E) :
    HasLineDerivAt 𝕜 f (L v) x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {x : E} {L : E →L[𝕜] F} (hf : HasFDerivAt f L x) (v : E)
Goal: HasLineDerivAt 𝕜 f (L v) x v
-- DifferentiableAt.lineDeriv_eq_fderiv
lemma DifferentiableAt.lineDeriv_eq_fderiv (hf : DifferentiableAt 𝕜 f x) :
    lineDeriv 𝕜 f x v = fderiv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {x v : E} (hf : DifferentiableAt 𝕜 f x)
Goal: lineDeriv 𝕜 f x v = (fderiv 𝕜 f x) v
-- LineDifferentiableWithinAt.mono_of_mem
theorem LineDifferentiableWithinAt.mono_of_mem (h : LineDifferentiableWithinAt 𝕜 f s x v)
    (hst : s ∈ 𝓝[t] x) : LineDifferentiableWithinAt 𝕜 f t x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s t : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (hst : s ∈ 𝓝[t] x)
Goal: LineDifferentiableWithinAt 𝕜 f t x v
-- lineDerivWithin_of_mem_nhds
theorem lineDerivWithin_of_mem_nhds (h : s ∈ 𝓝 x) :
    lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s : Set E} {x v : E} (h : s ∈ 𝓝 x)
Goal: lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v
-- lineDerivWithin_of_isOpen
theorem lineDerivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) :
    lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s : Set E} {x v : E} (hs : IsOpen s) (hx : x ∈ s)
Goal: lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v
-- hasLineDerivWithinAt_congr_set
theorem hasLineDerivWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
    HasLineDerivWithinAt 𝕜 f f' s x v ↔ HasLineDerivWithinAt 𝕜 f f' t x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {f' : F} {s t : Set E} {x v : E} (h : s =ᶠ[𝓝 x] t)
Goal: HasLineDerivWithinAt 𝕜 f f' s x v ↔ HasLineDerivWithinAt 𝕜 f f' t x v
-- lineDifferentiableWithinAt_congr_set
theorem lineDifferentiableWithinAt_congr_set (h : s =ᶠ[𝓝 x] t) :
    LineDifferentiableWithinAt 𝕜 f s x v ↔ LineDifferentiableWithinAt 𝕜 f t x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s t : Set E} {x v : E} (h : s =ᶠ[𝓝 x] t)
Goal: LineDifferentiableWithinAt 𝕜 f s x v ↔ LineDifferentiableWithinAt 𝕜 f t x v
-- lineDerivWithin_congr_set
theorem lineDerivWithin_congr_set (h : s =ᶠ[𝓝 x] t) :
    lineDerivWithin 𝕜 f s x v = lineDerivWithin 𝕜 f t x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {s t : Set E} {x v : E} (h : s =ᶠ[𝓝 x] t)
Goal: lineDerivWithin 𝕜 f s x v = lineDerivWithin 𝕜 f t x v
-- Filter.EventuallyEq.hasLineDerivAt_iff
theorem Filter.EventuallyEq.hasLineDerivAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
    HasLineDerivAt 𝕜 f₀ f' x v ↔ HasLineDerivAt 𝕜 f₁ f' x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {f' : F} {x v : E} (h : f₀ =ᶠ[𝓝 x] f₁)
Goal: HasLineDerivAt 𝕜 f₀ f' x v ↔ HasLineDerivAt 𝕜 f₁ f' x v
-- Filter.EventuallyEq.lineDifferentiableAt_iff
theorem Filter.EventuallyEq.lineDifferentiableAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
    LineDifferentiableAt 𝕜 f₀ x v ↔ LineDifferentiableAt 𝕜 f₁ x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {x v : E} (h : f₀ =ᶠ[𝓝 x] f₁)
Goal: LineDifferentiableAt 𝕜 f₀ x v ↔ LineDifferentiableAt 𝕜 f₁ x v
-- Filter.EventuallyEq.hasLineDerivWithinAt_iff
theorem Filter.EventuallyEq.hasLineDerivWithinAt_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
    HasLineDerivWithinAt 𝕜 f₀ f' s x v ↔ HasLineDerivWithinAt 𝕜 f₁ f' s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {f' : F} {s : Set E} {x v : E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x)
Goal: HasLineDerivWithinAt 𝕜 f₀ f' s x v ↔ HasLineDerivWithinAt 𝕜 f₁ f' s x v
-- Filter.EventuallyEq.hasLineDerivWithinAt_iff_of_mem
theorem Filter.EventuallyEq.hasLineDerivWithinAt_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
    HasLineDerivWithinAt 𝕜 f₀ f' s x v ↔ HasLineDerivWithinAt 𝕜 f₁ f' s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {f' : F} {s : Set E} {x v : E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s)
Goal: HasLineDerivWithinAt 𝕜 f₀ f' s x v ↔ HasLineDerivWithinAt 𝕜 f₁ f' s x v
-- Filter.EventuallyEq.lineDifferentiableWithinAt_iff
theorem Filter.EventuallyEq.lineDifferentiableWithinAt_iff
    (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
    LineDifferentiableWithinAt 𝕜 f₀ s x v ↔ LineDifferentiableWithinAt 𝕜 f₁ s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {s : Set E} {x v : E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x)
Goal: LineDifferentiableWithinAt 𝕜 f₀ s x v ↔ LineDifferentiableWithinAt 𝕜 f₁ s x v
-- Filter.EventuallyEq.lineDifferentiableWithinAt_iff_of_mem
theorem Filter.EventuallyEq.lineDifferentiableWithinAt_iff_of_mem
    (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
    LineDifferentiableWithinAt 𝕜 f₀ s x v ↔ LineDifferentiableWithinAt 𝕜 f₁ s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f₀ f₁ : E → F} {s : Set E} {x v : E} (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s)
Goal: LineDifferentiableWithinAt 𝕜 f₀ s x v ↔ LineDifferentiableWithinAt 𝕜 f₁ s x v
-- HasLineDerivWithinAt.congr_of_eventuallyEq
lemma HasLineDerivWithinAt.congr_of_eventuallyEq (hf : HasLineDerivWithinAt 𝕜 f f' s x v)
    (h'f : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : HasLineDerivWithinAt 𝕜 f₁ f' s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {f' : F} {s : Set E} {x v : E} (hf : HasLineDerivWithinAt 𝕜 f f' s x v) (h'f : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: HasLineDerivWithinAt 𝕜 f₁ f' s x v
-- HasLineDerivAt.congr_of_eventuallyEq
theorem HasLineDerivAt.congr_of_eventuallyEq (h : HasLineDerivAt 𝕜 f f' x v) (h₁ : f₁ =ᶠ[𝓝 x] f) :
    HasLineDerivAt 𝕜 f₁ f' x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {f' : F} {x v : E} (h : HasLineDerivAt 𝕜 f f' x v) (h₁ : f₁ =ᶠ[𝓝 x] f)
Goal: HasLineDerivAt 𝕜 f₁ f' x v
-- LineDifferentiableWithinAt.congr_of_eventuallyEq
theorem LineDifferentiableWithinAt.congr_of_eventuallyEq (h : LineDifferentiableWithinAt 𝕜 f s x v)
    (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : LineDifferentiableWithinAt 𝕜 f₁ s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: LineDifferentiableWithinAt 𝕜 f₁ s x v
-- LineDifferentiableAt.congr_of_eventuallyEq
theorem LineDifferentiableAt.congr_of_eventuallyEq
    (h : LineDifferentiableAt 𝕜 f x v) (hL : f₁ =ᶠ[𝓝 x] f) :
    LineDifferentiableAt 𝕜 f₁ x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {x v : E} (h : LineDifferentiableAt 𝕜 f x v) (hL : f₁ =ᶠ[𝓝 x] f)
Goal: LineDifferentiableAt 𝕜 f₁ x v
-- Filter.EventuallyEq.lineDerivWithin_eq
theorem Filter.EventuallyEq.lineDerivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
    lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
Goal: lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v
-- Filter.EventuallyEq.lineDerivWithin_eq_nhds
theorem Filter.EventuallyEq.lineDerivWithin_eq_nhds (h : f₁ =ᶠ[𝓝 x] f) :
    lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (h : f₁ =ᶠ[𝓝 x] f)
Goal: lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v
-- Filter.EventuallyEq.lineDeriv_eq
theorem Filter.EventuallyEq.lineDeriv_eq (h : f₁ =ᶠ[𝓝 x] f) :
    lineDeriv 𝕜 f₁ x v = lineDeriv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f f₁ : E → F} {x v : E} (h : f₁ =ᶠ[𝓝 x] f)
Goal: lineDeriv 𝕜 f₁ x v = lineDeriv 𝕜 f x v
-- HasLineDerivAt.le_of_lip'
theorem HasLineDerivAt.le_of_lip' {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v)
    {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
    ‖f'‖ ≤ C * ‖v‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v) {C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖)
Goal: ‖f'‖ ≤ C * ‖v‖
-- HasLineDerivAt.le_of_lipschitzOn
theorem HasLineDerivAt.le_of_lipschitzOn
    {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v)
    {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s) :
    ‖f'‖ ≤ C * ‖v‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v) {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖f'‖ ≤ ↑C * ‖v‖
-- HasLineDerivAt.le_of_lipschitz
theorem HasLineDerivAt.le_of_lipschitz
    {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v)
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖f'‖ ≤ C * ‖v‖ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v) {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖f'‖ ≤ ↑C * ‖v‖
-- norm_lineDeriv_le_of_lip'
theorem norm_lineDeriv_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‖) :
    ‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {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: ‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖
-- norm_lineDeriv_le_of_lipschitzOn
theorem norm_lineDeriv_le_of_lipschitzOn {f : E → F} {x₀ : E} {s : Set E} (hs : s ∈ 𝓝 x₀)
    {C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {f : E → F} {x₀ : E} {s : Set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : LipschitzOnWith C f s)
Goal: ‖lineDeriv 𝕜 f x₀ v‖ ≤ ↑C * ‖v‖
-- norm_lineDeriv_le_of_lipschitz
theorem norm_lineDeriv_le_of_lipschitz {f : E → F} {x₀ : E}
    {C : ℝ≥0} (hlip : LipschitzWith C f) : ‖lineDeriv 𝕜 f x₀ v‖ ≤ C * ‖v‖ 
Arguments: (𝕜 : Type u_1) [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {v : E} {f : E → F} {x₀ : E} {C : ℝ≥0} (hlip : LipschitzWith C f)
Goal: ‖lineDeriv 𝕜 f x₀ v‖ ≤ ↑C * ‖v‖
-- hasLineDerivWithinAt_zero
theorem hasLineDerivWithinAt_zero : HasLineDerivWithinAt 𝕜 f 0 s x 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x : E}
Goal: HasLineDerivWithinAt 𝕜 f 0 s x 0
-- hasLineDerivAt_zero
theorem hasLineDerivAt_zero : HasLineDerivAt 𝕜 f 0 x 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x : E}
Goal: HasLineDerivAt 𝕜 f 0 x 0
-- lineDifferentiableWithinAt_zero
theorem lineDifferentiableWithinAt_zero : LineDifferentiableWithinAt 𝕜 f s x 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x : E}
Goal: LineDifferentiableWithinAt 𝕜 f s x 0
-- lineDifferentiableAt_zero
theorem lineDifferentiableAt_zero : LineDifferentiableAt 𝕜 f x 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x : E}
Goal: LineDifferentiableAt 𝕜 f x 0
-- lineDeriv_zero
theorem lineDeriv_zero : lineDeriv 𝕜 f x 0 = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x : E}
Goal: lineDeriv 𝕜 f x 0 = 0
-- HasLineDerivAt.of_comp
theorem HasLineDerivAt.of_comp {v : E'} (hf : HasLineDerivAt 𝕜 (f ∘ L) f' x v) :
    HasLineDerivAt 𝕜 f f' (L x) (L v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {E' : Type u_4} [AddCommGroup E'] [Module 𝕜 E'] {f : E → F} {f' : F} {x : E'} {L : E' →ₗ[𝕜] E} {v : E'} (hf : HasLineDerivAt 𝕜 (f ∘ ⇑L) f' x v)
Goal: HasLineDerivAt 𝕜 f f' (L x) (L v)
-- LineDifferentiableAt.of_comp
theorem LineDifferentiableAt.of_comp {v : E'} (hf : LineDifferentiableAt 𝕜 (f ∘ L) x v) :
    LineDifferentiableAt 𝕜 f (L x) (L v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {E' : Type u_4} [AddCommGroup E'] [Module 𝕜 E'] {f : E → F} {x : E'} {L : E' →ₗ[𝕜] E} {v : E'} (hf : LineDifferentiableAt 𝕜 (f ∘ ⇑L) x v)
Goal: LineDifferentiableAt 𝕜 f (L x) (L v)
-- HasLineDerivWithinAt.smul
theorem HasLineDerivWithinAt.smul (h : HasLineDerivWithinAt 𝕜 f f' s x v) (c : 𝕜) :
    HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x v : E} {f' : F} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (c : 𝕜)
Goal: HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v)
-- hasLineDerivWithinAt_smul_iff
theorem hasLineDerivWithinAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
    HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v) ↔ HasLineDerivWithinAt 𝕜 f f' s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x v : E} {f' : F} {c : 𝕜} (hc : c ≠ 0)
Goal: HasLineDerivWithinAt 𝕜 f (c • f') s x (c • v) ↔ HasLineDerivWithinAt 𝕜 f f' s x v
-- HasLineDerivAt.smul
theorem HasLineDerivAt.smul (h : HasLineDerivAt 𝕜 f f' x v) (c : 𝕜) :
    HasLineDerivAt 𝕜 f (c • f') x (c • v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E} {f' : F} (h : HasLineDerivAt 𝕜 f f' x v) (c : 𝕜)
Goal: HasLineDerivAt 𝕜 f (c • f') x (c • v)
-- hasLineDerivAt_smul_iff
theorem hasLineDerivAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
    HasLineDerivAt 𝕜 f (c • f') x (c • v) ↔ HasLineDerivAt 𝕜 f f' x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E} {f' : F} {c : 𝕜} (hc : c ≠ 0)
Goal: HasLineDerivAt 𝕜 f (c • f') x (c • v) ↔ HasLineDerivAt 𝕜 f f' x v
-- LineDifferentiableWithinAt.smul
theorem LineDifferentiableWithinAt.smul (h : LineDifferentiableWithinAt 𝕜 f s x v) (c : 𝕜) :
    LineDifferentiableWithinAt 𝕜 f s x (c • v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (c : 𝕜)
Goal: LineDifferentiableWithinAt 𝕜 f s x (c • v)
-- lineDifferentiableWithinAt_smul_iff
theorem lineDifferentiableWithinAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
    LineDifferentiableWithinAt 𝕜 f s x (c • v) ↔ LineDifferentiableWithinAt 𝕜 f s x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {s : Set E} {x v : E} {c : 𝕜} (hc : c ≠ 0)
Goal: LineDifferentiableWithinAt 𝕜 f s x (c • v) ↔ LineDifferentiableWithinAt 𝕜 f s x v
-- LineDifferentiableAt.smul
theorem LineDifferentiableAt.smul (h : LineDifferentiableAt 𝕜 f x v) (c : 𝕜) :
    LineDifferentiableAt 𝕜 f x (c • v) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E} (h : LineDifferentiableAt 𝕜 f x v) (c : 𝕜)
Goal: LineDifferentiableAt 𝕜 f x (c • v)
-- lineDifferentiableAt_smul_iff
theorem lineDifferentiableAt_smul_iff {c : 𝕜} (hc : c ≠ 0) :
    LineDifferentiableAt 𝕜 f x (c • v) ↔ LineDifferentiableAt 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E} {c : 𝕜} (hc : c ≠ 0)
Goal: LineDifferentiableAt 𝕜 f x (c • v) ↔ LineDifferentiableAt 𝕜 f x v
-- lineDeriv_smul
theorem lineDeriv_smul {c : 𝕜} : lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E} {c : 𝕜}
Goal: lineDeriv 𝕜 f x (c • v) = c • lineDeriv 𝕜 f x v
-- lineDeriv_neg
theorem lineDeriv_neg : lineDeriv 𝕜 f x (-v) = - lineDeriv 𝕜 f x v 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {x v : E}
Goal: lineDeriv 𝕜 f x (-v) = -lineDeriv 𝕜 f x v
-- ContinuousLinearMap.hasStrictFDerivAt
@[fun_prop]
protected theorem ContinuousLinearMap.hasStrictFDerivAt {x : E} : HasStrictFDerivAt e e 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] (e : E →L[𝕜] F) {x : E}
Goal: HasStrictFDerivAt (⇑e) e x
-- ContinuousLinearMap.hasFDerivAtFilter
protected theorem ContinuousLinearMap.hasFDerivAtFilter : HasFDerivAtFilter e e 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] (e : E →L[𝕜] F) {x : E} {L : Filter E}
Goal: HasFDerivAtFilter (⇑e) e x L
-- ContinuousLinearMap.hasFDerivWithinAt
@[fun_prop]
protected theorem ContinuousLinearMap.hasFDerivWithinAt : HasFDerivWithinAt e e 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] (e : E →L[𝕜] F) {x : E} {s : Set E}
Goal: HasFDerivWithinAt (⇑e) e s x
-- ContinuousLinearMap.hasFDerivAt
@[fun_prop]
protected theorem ContinuousLinearMap.hasFDerivAt : HasFDerivAt e e 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] (e : E →L[𝕜] F) {x : E}
Goal: HasFDerivAt (⇑e) e x
-- ContinuousLinearMap.differentiableAt
@[simp, fun_prop]
protected theorem ContinuousLinearMap.differentiableAt : DifferentiableAt 𝕜 e 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] (e : E →L[𝕜] F) {x : E}
Goal: DifferentiableAt 𝕜 (⇑e) x
-- ContinuousLinearMap.differentiableWithinAt
@[fun_prop]
protected theorem ContinuousLinearMap.differentiableWithinAt : DifferentiableWithinAt 𝕜 e 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] (e : E →L[𝕜] F) {x : E} {s : Set E}
Goal: DifferentiableWithinAt 𝕜 (⇑e) s x
-- ContinuousLinearMap.fderiv
@[simp]
protected theorem ContinuousLinearMap.fderiv : fderiv 𝕜 e x = e 
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] (e : E →L[𝕜] F) {x : E}
Goal: fderiv 𝕜 (⇑e) x = e
-- ContinuousLinearMap.fderivWithin
protected theorem ContinuousLinearMap.fderivWithin (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 e s x = e 
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] (e : E →L[𝕜] F) {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (⇑e) s x = e
-- ContinuousLinearMap.differentiable
@[simp, fun_prop]
protected theorem ContinuousLinearMap.differentiable : Differentiable 𝕜 e 
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] (e : E →L[𝕜] F)
Goal: Differentiable 𝕜 ⇑e