ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq
theorem eLpNorm_le_eLpNorm_fderiv_of_eq [FiniteDimensional ℝ F]
    {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
    {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
    (hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
    eLpNorm u p' μ ≤ SNormLESNormFDerivOfEqConst F μ p * eLpNorm (fderiv ℝ u) p μ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {u : E → F} (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u) {p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < FiniteDimensional.finrank ℝ E) (hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑(FiniteDimensional.finrank ℝ E))⁻¹)
Goal: MeasureTheory.eLpNorm u (↑p') μ ≤
  ↑(MeasureTheory.SNormLESNormFDerivOfEqConst F μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ
-- MeasureTheory.eLpNormLESNormFDerivOfLeConst_def
irreducible_def eLpNormLESNormFDerivOfLeConst [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0) :
    ℝ≥0 
Arguments: (F : Type u_6) [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_7} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] (s : Set E) (p q : ℝ≥0)
Goal: MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p q =
  let p' := (p⁻¹ - (↑(FiniteDimensional.finrank ℝ E))⁻¹)⁻¹;
  (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p') * MeasureTheory.SNormLESNormFDerivOfEqConst F μ ↑p
-- MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_le
theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
    {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : u.support ⊆ s)
    {p q : ℝ≥0} (hp : 1 ≤ p) (h2p : p < finrank ℝ E)
    (hpq : p⁻¹ - (finrank ℝ E : ℝ)⁻¹ ≤ (q : ℝ)⁻¹)
    (hs : Bornology.IsBounded s) :
    eLpNorm u q μ ≤ eLpNormLESNormFDerivOfLeConst F μ s p q * eLpNorm (fderiv ℝ u) p μ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : Function.support u ⊆ s) {p q : ℝ≥0} (hp : 1 ≤ p) (h2p : p < ↑(FiniteDimensional.finrank ℝ E)) (hpq : ↑p⁻¹ - (↑(FiniteDimensional.finrank ℝ E))⁻¹ ≤ (↑q)⁻¹) (hs : Bornology.IsBounded s)
Goal: MeasureTheory.eLpNorm u (↑q) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p q) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ
-- MeasureTheory.eLpNorm_le_eLpNorm_fderiv
theorem eLpNorm_le_eLpNorm_fderiv [FiniteDimensional ℝ F]
    {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : u.support ⊆ s)
    {p : ℝ≥0} (hp : 1 ≤ p) (h2p : p < finrank ℝ E) (hs : Bornology.IsBounded s) :
    eLpNorm u p μ ≤ eLpNormLESNormFDerivOfLeConst F μ s p p * eLpNorm (fderiv ℝ u) p μ 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] [FiniteDimensional ℝ F] {u : E → F} {s : Set E} (hu : ContDiff ℝ 1 u) (h2u : Function.support u ⊆ s) {p : ℝ≥0} (hp : 1 ≤ p) (h2p : p < ↑(FiniteDimensional.finrank ℝ E)) (hs : Bornology.IsBounded s)
Goal: MeasureTheory.eLpNorm u (↑p) μ ≤
  ↑(MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p p) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ
-- HasStrictFDerivAt.star
@[fun_prop]
theorem HasStrictFDerivAt.star (h : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (fun x => star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (fun x => star (f x)) ((↑(starL' 𝕜)).comp f') x
-- HasFDerivAtFilter.star
theorem HasFDerivAtFilter.star (h : HasFDerivAtFilter f f' x L) :
    HasFDerivAtFilter (fun x => star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x L 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L : Filter E} (h : HasFDerivAtFilter f f' x L)
Goal: HasFDerivAtFilter (fun x => star (f x)) ((↑(starL' 𝕜)).comp f') x L
-- HasFDerivWithinAt.star
@[fun_prop]
nonrec theorem HasFDerivWithinAt.star (h : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (fun x => star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E} (h : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (fun x => star (f x)) ((↑(starL' 𝕜)).comp f') s x
-- HasFDerivAt.star
@[fun_prop]
nonrec theorem HasFDerivAt.star (h : HasFDerivAt f f' x) :
    HasFDerivAt (fun x => star (f x)) (((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L f') x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (h : HasFDerivAt f f' x)
Goal: HasFDerivAt (fun x => star (f x)) ((↑(starL' 𝕜)).comp f') x
-- DifferentiableWithinAt.star
@[fun_prop]
theorem DifferentiableWithinAt.star (h : DifferentiableWithinAt 𝕜 f s x) :
    DifferentiableWithinAt 𝕜 (fun y => star (f y)) s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E} {s : Set E} (h : DifferentiableWithinAt 𝕜 f s x)
Goal: DifferentiableWithinAt 𝕜 (fun y => star (f y)) s x
-- differentiableWithinAt_star_iff
@[simp]
theorem differentiableWithinAt_star_iff :
    DifferentiableWithinAt 𝕜 (fun y => star (f y)) s x ↔ DifferentiableWithinAt 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E} {s : Set E}
Goal: DifferentiableWithinAt 𝕜 (fun y => star (f y)) s x ↔ DifferentiableWithinAt 𝕜 f s x
-- DifferentiableAt.star
@[fun_prop]
theorem DifferentiableAt.star (h : DifferentiableAt 𝕜 f x) :
    DifferentiableAt 𝕜 (fun y => star (f y)) x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E} (h : DifferentiableAt 𝕜 f x)
Goal: DifferentiableAt 𝕜 (fun y => star (f y)) x
-- differentiableAt_star_iff
@[simp]
theorem differentiableAt_star_iff :
    DifferentiableAt 𝕜 (fun y => star (f y)) x ↔ DifferentiableAt 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E}
Goal: DifferentiableAt 𝕜 (fun y => star (f y)) x ↔ DifferentiableAt 𝕜 f x
-- DifferentiableOn.star
@[fun_prop]
theorem DifferentiableOn.star (h : DifferentiableOn 𝕜 f s) :
    DifferentiableOn 𝕜 (fun y => star (f y)) s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {s : Set E} (h : DifferentiableOn 𝕜 f s)
Goal: DifferentiableOn 𝕜 (fun y => star (f y)) s
-- differentiableOn_star_iff
@[simp]
theorem differentiableOn_star_iff :
    DifferentiableOn 𝕜 (fun y => star (f y)) s ↔ DifferentiableOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {s : Set E}
Goal: DifferentiableOn 𝕜 (fun y => star (f y)) s ↔ DifferentiableOn 𝕜 f s
-- Differentiable.star
@[fun_prop]
theorem Differentiable.star (h : Differentiable 𝕜 f) : Differentiable 𝕜 fun y => star (f y) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} (h : Differentiable 𝕜 f)
Goal: Differentiable 𝕜 fun y => star (f y)
-- differentiable_star_iff
@[simp]
theorem differentiable_star_iff : (Differentiable 𝕜 fun y => star (f y)) ↔ Differentiable 𝕜 f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F}
Goal: (Differentiable 𝕜 fun y => star (f y)) ↔ Differentiable 𝕜 f
-- fderivWithin_star
theorem fderivWithin_star (hxs : UniqueDiffWithinAt 𝕜 s x) :
    fderivWithin 𝕜 (fun y => star (f y)) s x =
      ((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L fderivWithin 𝕜 f s x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E} {s : Set E} (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: fderivWithin 𝕜 (fun y => star (f y)) s x = (↑(starL' 𝕜)).comp (fderivWithin 𝕜 f s x)
-- fderiv_star
@[simp]
theorem fderiv_star :
    fderiv 𝕜 (fun y => star (f y)) x = ((starL' 𝕜 : F ≃L[𝕜] F) : F →L[𝕜] F) ∘L fderiv 𝕜 f x 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] [StarRing 𝕜] [TrivialStar 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [StarAddMonoid F] [NormedSpace 𝕜 F] [StarModule 𝕜 F] [ContinuousStar F] {f : E → F} {x : E}
Goal: fderiv 𝕜 (fun y => star (f y)) x = (↑(starL' 𝕜)).comp (fderiv 𝕜 f x)
-- HasLineDerivWithinAt.mono
lemma HasLineDerivWithinAt.mono (hf : HasLineDerivWithinAt 𝕜 f f' s x v) (hst : t ⊆ s) :
    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} [AddCommGroup E] [Module 𝕜 E] {f : E → F} {f' : F} {s t : Set E} {x v : E} (hf : HasLineDerivWithinAt 𝕜 f f' s x v) (hst : t ⊆ s)
Goal: HasLineDerivWithinAt 𝕜 f f' t x v
-- HasLineDerivAt.hasLineDerivWithinAt
lemma HasLineDerivAt.hasLineDerivWithinAt (hf : HasLineDerivAt 𝕜 f f' x v) (s : Set E) :
    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} {f' : F} {x v : E} (hf : HasLineDerivAt 𝕜 f f' x v) (s : Set E)
Goal: HasLineDerivWithinAt 𝕜 f f' s x v
-- HasLineDerivWithinAt.lineDifferentiableWithinAt
lemma HasLineDerivWithinAt.lineDifferentiableWithinAt (hf : HasLineDerivWithinAt 𝕜 f f' s x 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} {f' : F} {s : Set E} {x v : E} (hf : HasLineDerivWithinAt 𝕜 f f' s x v)
Goal: LineDifferentiableWithinAt 𝕜 f s x v
-- HasLineDerivAt.lineDifferentiableAt
theorem HasLineDerivAt.lineDifferentiableAt (hf : HasLineDerivAt 𝕜 f f' x 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} {f' : F} {x v : E} (hf : HasLineDerivAt 𝕜 f f' x v)
Goal: LineDifferentiableAt 𝕜 f x v
-- LineDifferentiableWithinAt.hasLineDerivWithinAt
theorem LineDifferentiableWithinAt.hasLineDerivWithinAt (h : LineDifferentiableWithinAt 𝕜 f s x v) :
    HasLineDerivWithinAt 𝕜 f (lineDerivWithin 𝕜 f s x v) 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} (h : LineDifferentiableWithinAt 𝕜 f s x v)
Goal: HasLineDerivWithinAt 𝕜 f (lineDerivWithin 𝕜 f s x v) s x v
-- LineDifferentiableAt.hasLineDerivAt
theorem LineDifferentiableAt.hasLineDerivAt (h : LineDifferentiableAt 𝕜 f x v) :
    HasLineDerivAt 𝕜 f (lineDeriv 𝕜 f x v) 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} (h : LineDifferentiableAt 𝕜 f x v)
Goal: HasLineDerivAt 𝕜 f (lineDeriv 𝕜 f x v) x v
-- hasLineDerivWithinAt_univ
@[simp] lemma hasLineDerivWithinAt_univ :
    HasLineDerivWithinAt 𝕜 f f' univ x 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} {f' : F} {x v : E}
Goal: HasLineDerivWithinAt 𝕜 f f' Set.univ x v ↔ HasLineDerivAt 𝕜 f f' x v
-- lineDerivWithin_zero_of_not_lineDifferentiableWithinAt
theorem lineDerivWithin_zero_of_not_lineDifferentiableWithinAt
    (h : ¬LineDifferentiableWithinAt 𝕜 f s x v) :
    lineDerivWithin 𝕜 f s x v = 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 v : E} (h : ¬LineDifferentiableWithinAt 𝕜 f s x v)
Goal: lineDerivWithin 𝕜 f s x v = 0
-- lineDeriv_zero_of_not_lineDifferentiableAt
theorem lineDeriv_zero_of_not_lineDifferentiableAt (h : ¬LineDifferentiableAt 𝕜 f x v) :
    lineDeriv 𝕜 f x v = 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 v : E} (h : ¬LineDifferentiableAt 𝕜 f x v)
Goal: lineDeriv 𝕜 f x v = 0
-- hasLineDerivAt_iff_isLittleO_nhds_zero
theorem hasLineDerivAt_iff_isLittleO_nhds_zero :
    HasLineDerivAt 𝕜 f f' x v ↔
      (fun t : 𝕜 => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t 
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} {f' : F} {x v : E}
Goal: HasLineDerivAt 𝕜 f f' x v ↔ (fun t => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t
-- HasLineDerivAt.unique
theorem HasLineDerivAt.unique (h₀ : HasLineDerivAt 𝕜 f f₀' x v) (h₁ : HasLineDerivAt 𝕜 f f₁' x v) :
    f₀' = f₁' 
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} {f₀' f₁' : F} {x v : E} (h₀ : HasLineDerivAt 𝕜 f f₀' x v) (h₁ : HasLineDerivAt 𝕜 f f₁' x v)
Goal: f₀' = f₁'
-- HasLineDerivAt.lineDeriv
protected theorem HasLineDerivAt.lineDeriv (h : HasLineDerivAt 𝕜 f f' x v) :
    lineDeriv 𝕜 f x v = f' 
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} {f' : F} {x v : E} (h : HasLineDerivAt 𝕜 f f' x v)
Goal: lineDeriv 𝕜 f x v = f'
-- lineDifferentiableWithinAt_univ
theorem lineDifferentiableWithinAt_univ :
    LineDifferentiableWithinAt 𝕜 f univ x 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}
Goal: LineDifferentiableWithinAt 𝕜 f Set.univ x v ↔ LineDifferentiableAt 𝕜 f x v
-- LineDifferentiableAt.lineDifferentiableWithinAt
theorem LineDifferentiableAt.lineDifferentiableWithinAt (h : LineDifferentiableAt 𝕜 f x 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} (h : LineDifferentiableAt 𝕜 f x v)
Goal: LineDifferentiableWithinAt 𝕜 f s x v
-- lineDerivWithin_univ
@[simp]
theorem lineDerivWithin_univ : lineDerivWithin 𝕜 f univ 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: lineDerivWithin 𝕜 f Set.univ x v = lineDeriv 𝕜 f x v
-- LineDifferentiableWithinAt.mono
theorem LineDifferentiableWithinAt.mono (h : LineDifferentiableWithinAt 𝕜 f t x v) (st : s ⊆ t) :
    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 t : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f t x v) (st : s ⊆ t)
Goal: LineDifferentiableWithinAt 𝕜 f s x v
-- HasLineDerivWithinAt.congr_mono
theorem HasLineDerivWithinAt.congr_mono (h : HasLineDerivWithinAt 𝕜 f f' s x v) (ht : EqOn f₁ f t)
    (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasLineDerivWithinAt 𝕜 f₁ f' t 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 f₁ : E → F} {f' : F} {s t : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (ht : Set.EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: HasLineDerivWithinAt 𝕜 f₁ f' t x v
-- HasLineDerivWithinAt.congr
theorem HasLineDerivWithinAt.congr (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : EqOn f₁ f s)
    (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} [AddCommGroup E] [Module 𝕜 E] {f f₁ : E → F} {f' : F} {s : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : Set.EqOn f₁ f s) (hx : f₁ x = f x)
Goal: HasLineDerivWithinAt 𝕜 f₁ f' s x v
-- HasLineDerivWithinAt.congr'
theorem HasLineDerivWithinAt.congr' (h : HasLineDerivWithinAt 𝕜 f f' s x v)
    (hs : EqOn f₁ f s) (hx : x ∈ s) :
    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} [AddCommGroup E] [Module 𝕜 E] {f f₁ : E → F} {f' : F} {s : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : Set.EqOn f₁ f s) (hx : x ∈ s)
Goal: HasLineDerivWithinAt 𝕜 f₁ f' s x v
-- LineDifferentiableWithinAt.congr_mono
theorem LineDifferentiableWithinAt.congr_mono (h : LineDifferentiableWithinAt 𝕜 f s x v)
    (ht : EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s) :
    LineDifferentiableWithinAt 𝕜 f₁ t 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 f₁ : E → F} {s t : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (ht : Set.EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s)
Goal: LineDifferentiableWithinAt 𝕜 f₁ t x v
-- LineDifferentiableWithinAt.congr
theorem LineDifferentiableWithinAt.congr (h : LineDifferentiableWithinAt 𝕜 f s x v)
    (ht : ∀ x ∈ s, f₁ x = f x) (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} [AddCommGroup E] [Module 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (h : LineDifferentiableWithinAt 𝕜 f s x v) (ht : ∀ x ∈ s, f₁ x = f x) (hx : f₁ x = f x)
Goal: LineDifferentiableWithinAt 𝕜 f₁ s x v
-- lineDerivWithin_congr
theorem lineDerivWithin_congr (hs : EqOn f₁ f s) (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} [AddCommGroup E] [Module 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (hs : Set.EqOn f₁ f s) (hx : f₁ x = f x)
Goal: lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v
-- lineDerivWithin_congr'
theorem lineDerivWithin_congr' (hs : EqOn f₁ f s) (hx : x ∈ s) :
    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} [AddCommGroup E] [Module 𝕜 E] {f f₁ : E → F} {s : Set E} {x v : E} (hs : Set.EqOn f₁ f s) (hx : x ∈ s)
Goal: lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v
-- hasLineDerivAt_iff_tendsto_slope_zero
theorem hasLineDerivAt_iff_tendsto_slope_zero :
    HasLineDerivAt 𝕜 f f' x v ↔
      Tendsto (fun (t : 𝕜) ↦ t⁻¹ • (f (x + t • v) - f x)) (𝓝[≠] 0) (𝓝 f') 
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} {f' : F} {x v : E}
Goal: HasLineDerivAt 𝕜 f f' x v ↔ Filter.Tendsto (fun t => t⁻¹ • (f (x + t • v) - f x)) (𝓝[≠] 0) (𝓝 f')
-- HasLineDerivAt.tendsto_slope_zero_right
theorem HasLineDerivAt.tendsto_slope_zero_right [PartialOrder 𝕜] (h : HasLineDerivAt 𝕜 f f' x v) :
    Tendsto (fun (t : 𝕜) ↦ t⁻¹ • (f (x + t • v) - f x)) (𝓝[>] 0) (𝓝 f') 
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} {f' : F} {x v : E} [PartialOrder 𝕜] (h : HasLineDerivAt 𝕜 f f' x v)
Goal: Filter.Tendsto (fun t => t⁻¹ • (f (x + t • v) - f x)) (𝓝[>] 0) (𝓝 f')
-- HasLineDerivAt.tendsto_slope_zero_left
theorem HasLineDerivAt.tendsto_slope_zero_left [PartialOrder 𝕜] (h : HasLineDerivAt 𝕜 f f' x v) :
    Tendsto (fun (t : 𝕜) ↦ t⁻¹ • (f (x + t • v) - f x)) (𝓝[<] 0) (𝓝 f') 
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} {f' : F} {x v : E} [PartialOrder 𝕜] (h : HasLineDerivAt 𝕜 f f' x v)
Goal: Filter.Tendsto (fun t => t⁻¹ • (f (x + t • v) - f x)) (𝓝[<] 0) (𝓝 f')
-- HasLineDerivWithinAt.hasLineDerivAt'
theorem HasLineDerivWithinAt.hasLineDerivAt'
    (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : ∀ᶠ t : 𝕜 in 𝓝 0, x + t • v ∈ s) :
    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} {f' : F} {s : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : ∀ᶠ (t : 𝕜) in 𝓝 0, x + t • v ∈ s)
Goal: HasLineDerivAt 𝕜 f f' x v
-- HasLineDerivWithinAt.mono_of_mem
theorem HasLineDerivWithinAt.mono_of_mem
    (h : HasLineDerivWithinAt 𝕜 f f' t x v) (hst : t ∈ 𝓝[s] x) :
    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} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {f' : F} {s t : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' t x v) (hst : t ∈ 𝓝[s] x)
Goal: HasLineDerivWithinAt 𝕜 f f' s x v
-- HasLineDerivWithinAt.hasLineDerivAt
theorem HasLineDerivWithinAt.hasLineDerivAt
    (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : s ∈ 𝓝 x) :
    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} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → F} {f' : F} {s : Set E} {x v : E} (h : HasLineDerivWithinAt 𝕜 f f' s x v) (hs : s ∈ 𝓝 x)
Goal: HasLineDerivAt 𝕜 f f' x v