ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsExtrOn.lineDerivWithin_eq_zero
theorem IsExtrOn.lineDerivWithin_eq_zero (h : IsExtrOn f s a)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : lineDerivWithin ℝ f s a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsExtrOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDerivWithin ℝ f s a b = 0
-- IsMinOn.hasLineDerivWithinAt_eq_zero
theorem IsMinOn.hasLineDerivWithinAt_eq_zero (h : IsMinOn f s a)
    (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsMinOn f s a) (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0
-- IsMinOn.lineDerivWithin_eq_zero
theorem IsMinOn.lineDerivWithin_eq_zero (h : IsMinOn f s a)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : lineDerivWithin ℝ f s a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsMinOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDerivWithin ℝ f s a b = 0
-- IsMaxOn.hasLineDerivWithinAt_eq_zero
theorem IsMaxOn.hasLineDerivWithinAt_eq_zero (h : IsMaxOn f s a)
    (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} {f' : ℝ} (h : IsMaxOn f s a) (hd : HasLineDerivWithinAt ℝ f f' s a b) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: f' = 0
-- IsMaxOn.lineDerivWithin_eq_zero
theorem IsMaxOn.lineDerivWithin_eq_zero (h : IsMaxOn f s a)
    (h' : ∀ᶠ t : ℝ in 𝓝 0, a + t • b ∈ s) : lineDerivWithin ℝ f s a b = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] {f : E → ℝ} {s : Set E} {a b : E} (h : IsMaxOn f s a) (h' : ∀ᶠ (t : ℝ) in 𝓝 0, a + t • b ∈ s)
Goal: lineDerivWithin ℝ f s a b = 0
-- IsLocalExtr.hasLineDerivAt_eq_zero
theorem IsLocalExtr.hasLineDerivAt_eq_zero (h : IsLocalExtr f a) (hd : HasLineDerivAt ℝ f f' a b) :
    f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} (h : IsLocalExtr f a) (hd : HasLineDerivAt ℝ f f' a b)
Goal: f' = 0
-- IsLocalExtr.lineDeriv_eq_zero
theorem IsLocalExtr.lineDeriv_eq_zero (h : IsLocalExtr f a) : lineDeriv ℝ f a = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a : E} (h : IsLocalExtr f a)
Goal: lineDeriv ℝ f a = 0
-- IsLocalMin.hasLineDerivAt_eq_zero
theorem IsLocalMin.hasLineDerivAt_eq_zero (h : IsLocalMin f a) (hd : HasLineDerivAt ℝ f f' a b) :
    f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} (h : IsLocalMin f a) (hd : HasLineDerivAt ℝ f f' a b)
Goal: f' = 0
-- IsLocalMin.lineDeriv_eq_zero
theorem IsLocalMin.lineDeriv_eq_zero (h : IsLocalMin f a) : lineDeriv ℝ f a = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a : E} (h : IsLocalMin f a)
Goal: lineDeriv ℝ f a = 0
-- IsLocalMax.hasLineDerivAt_eq_zero
theorem IsLocalMax.hasLineDerivAt_eq_zero (h : IsLocalMax f a) (hd : HasLineDerivAt ℝ f f' a b) :
    f' = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a b : E} {f' : ℝ} (h : IsLocalMax f a) (hd : HasLineDerivAt ℝ f f' a b)
Goal: f' = 0
-- IsLocalMax.lineDeriv_eq_zero
theorem IsLocalMax.lineDeriv_eq_zero (h : IsLocalMax f a) : lineDeriv ℝ f a = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {f : E → ℝ} {a : E} (h : IsLocalMax f a)
Goal: lineDeriv ℝ f a = 0
-- HasDerivAtFilter.scomp
theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter g₁ g₁' (h x) L')
    (hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
    HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {L : Filter 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {L' : Filter 𝕜'} (hg : HasDerivAtFilter g₁ g₁' (h x) L') (hh : HasDerivAtFilter h h' x L) (hL : Filter.Tendsto h L L')
Goal: HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L
-- HasDerivAtFilter.scomp_of_eq
theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter g₁ g₁' y L')
    (hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
    HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {L : Filter 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {L' : Filter 𝕜'} {y : 𝕜'} (hg : HasDerivAtFilter g₁ g₁' y L') (hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Filter.Tendsto h L L')
Goal: HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L
-- HasDerivWithinAt.scomp_hasDerivAt
theorem HasDerivWithinAt.scomp_hasDerivAt (hg : HasDerivWithinAt g₁ g₁' s' (h x))
    (hh : HasDerivAt h h' x) (hs : ∀ x, h x ∈ s') : HasDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} (hg : HasDerivWithinAt g₁ g₁' s' (h x)) (hh : HasDerivAt h h' x) (hs : ∀ (x : 𝕜), h x ∈ s')
Goal: HasDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasDerivWithinAt.scomp_hasDerivAt_of_eq
theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt g₁ g₁' s' y)
    (hh : HasDerivAt h h' x) (hs : ∀ x, h x ∈ s') (hy : y = h x) :
    HasDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {y : 𝕜'} (hg : HasDerivWithinAt g₁ g₁' s' y) (hh : HasDerivAt h h' x) (hs : ∀ (x : 𝕜), h x ∈ s') (hy : y = h x)
Goal: HasDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasDerivWithinAt.scomp
nonrec theorem HasDerivWithinAt.scomp (hg : HasDerivWithinAt g₁ g₁' t' (h x))
    (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') :
    HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {t' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} (hg : HasDerivWithinAt g₁ g₁' t' (h x)) (hh : HasDerivWithinAt h h' s x) (hst : Set.MapsTo h s t')
Goal: HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x
-- HasDerivWithinAt.scomp_of_eq
theorem HasDerivWithinAt.scomp_of_eq (hg : HasDerivWithinAt g₁ g₁' t' y)
    (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') (hy : y = h x) :
    HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {t' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {y : 𝕜'} (hg : HasDerivWithinAt g₁ g₁' t' y) (hh : HasDerivWithinAt h h' s x) (hst : Set.MapsTo h s t') (hy : y = h x)
Goal: HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x
-- HasDerivAt.scomp
nonrec theorem HasDerivAt.scomp (hg : HasDerivAt g₁ g₁' (h x)) (hh : HasDerivAt h h' x) :
    HasDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} (hg : HasDerivAt g₁ g₁' (h x)) (hh : HasDerivAt h h' x)
Goal: HasDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasDerivAt.scomp_of_eq
theorem HasDerivAt.scomp_of_eq
    (hg : HasDerivAt g₁ g₁' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
    HasDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {y : 𝕜'} (hg : HasDerivAt g₁ g₁' y) (hh : HasDerivAt h h' x) (hy : y = h x)
Goal: HasDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasStrictDerivAt.scomp
theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt g₁ g₁' (h x)) (hh : HasStrictDerivAt h h' x) :
    HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} (hg : HasStrictDerivAt g₁ g₁' (h x)) (hh : HasStrictDerivAt h h' x)
Goal: HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasStrictDerivAt.scomp_of_eq
theorem HasStrictDerivAt.scomp_of_eq
    (hg : HasStrictDerivAt g₁ g₁' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x) :
    HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {y : 𝕜'} (hg : HasStrictDerivAt g₁ g₁' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x)
Goal: HasStrictDerivAt (g₁ ∘ h) (h' • g₁') x
-- HasDerivAt.scomp_hasDerivWithinAt
theorem HasDerivAt.scomp_hasDerivWithinAt (hg : HasDerivAt g₁ g₁' (h x))
    (hh : HasDerivWithinAt h h' s x) : HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} (hg : HasDerivAt g₁ g₁' (h x)) (hh : HasDerivWithinAt h h' s x)
Goal: HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x
-- HasDerivAt.scomp_hasDerivWithinAt_of_eq
theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt g₁ g₁' y)
    (hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
    HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {h' : 𝕜'} {g₁ : 𝕜' → F} {g₁' : F} {y : 𝕜'} (hg : HasDerivAt g₁ g₁' y) (hh : HasDerivWithinAt h h' s x) (hy : y = h x)
Goal: HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x
-- derivWithin.scomp
theorem derivWithin.scomp (hg : DifferentiableWithinAt 𝕜' g₁ t' (h x))
    (hh : DifferentiableWithinAt 𝕜 h s x) (hs : MapsTo h s t') (hxs : UniqueDiffWithinAt 𝕜 s x) :
    derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {t' : Set 𝕜'} {h : 𝕜 → 𝕜'} {g₁ : 𝕜' → F} (hg : DifferentiableWithinAt 𝕜' g₁ t' (h x)) (hh : DifferentiableWithinAt 𝕜 h s x) (hs : Set.MapsTo h s t') (hxs : UniqueDiffWithinAt 𝕜 s x)
Goal: derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)
-- derivWithin.scomp_of_eq
theorem derivWithin.scomp_of_eq (hg : DifferentiableWithinAt 𝕜' g₁ t' y)
    (hh : DifferentiableWithinAt 𝕜 h s x) (hs : MapsTo h s t') (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hy : y = h x) :
    derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {t' : Set 𝕜'} {h : 𝕜 → 𝕜'} {g₁ : 𝕜' → F} {y : 𝕜'} (hg : DifferentiableWithinAt 𝕜' g₁ t' y) (hh : DifferentiableWithinAt 𝕜 h s x) (hs : Set.MapsTo h s t') (hxs : UniqueDiffWithinAt 𝕜 s x) (hy : y = h x)
Goal: derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x)
-- deriv.scomp
theorem deriv.scomp (hg : DifferentiableAt 𝕜' g₁ (h x)) (hh : DifferentiableAt 𝕜 h x) :
    deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {g₁ : 𝕜' → F} (hg : DifferentiableAt 𝕜' g₁ (h x)) (hh : DifferentiableAt 𝕜 h x)
Goal: deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x)
-- deriv.scomp_of_eq
theorem deriv.scomp_of_eq
    (hg : DifferentiableAt 𝕜' g₁ y) (hh : DifferentiableAt 𝕜 h x) (hy : y = h x) :
    deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {h : 𝕜 → 𝕜'} {g₁ : 𝕜' → F} {y : 𝕜'} (hg : DifferentiableAt 𝕜' g₁ y) (hh : DifferentiableAt 𝕜 h x) (hy : y = h x)
Goal: deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x)
-- HasDerivAtFilter.comp_hasFDerivAtFilter
theorem HasDerivAtFilter.comp_hasFDerivAtFilter {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x) {L'' : Filter E}
    (hh₂ : HasDerivAtFilter h₂ h₂' (f x) L') (hf : HasFDerivAtFilter f f' x L'')
    (hL : Tendsto f L'' L') : HasFDerivAtFilter (h₂ ∘ f) (h₂' • f') x L'' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {L' : Filter 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) {L'' : Filter E} (hh₂ : HasDerivAtFilter h₂ h₂' (f x) L') (hf : HasFDerivAtFilter f f' x L'') (hL : Filter.Tendsto f L'' L')
Goal: HasFDerivAtFilter (h₂ ∘ f) (h₂' • f') x L''
-- HasDerivAtFilter.comp_hasFDerivAtFilter_of_eq
theorem HasDerivAtFilter.comp_hasFDerivAtFilter_of_eq
    {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x) {L'' : Filter E}
    (hh₂ : HasDerivAtFilter h₂ h₂' y L') (hf : HasFDerivAtFilter f f' x L'')
    (hL : Tendsto f L'' L') (hy : y = f x) : HasFDerivAtFilter (h₂ ∘ f) (h₂' • f') x L'' 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {L' : Filter 𝕜'} {y : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) {L'' : Filter E} (hh₂ : HasDerivAtFilter h₂ h₂' y L') (hf : HasFDerivAtFilter f f' x L'') (hL : Filter.Tendsto f L'' L') (hy : y = f x)
Goal: HasFDerivAtFilter (h₂ ∘ f) (h₂' • f') x L''
-- HasStrictDerivAt.comp_hasStrictFDerivAt
theorem HasStrictDerivAt.comp_hasStrictFDerivAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
    (hh : HasStrictDerivAt h₂ h₂' (f x)) (hf : HasStrictFDerivAt f f' x) :
    HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) (hh : HasStrictDerivAt h₂ h₂' (f x)) (hf : HasStrictFDerivAt f f' x)
Goal: HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x
-- HasStrictDerivAt.comp_hasStrictFDerivAt_of_eq
theorem HasStrictDerivAt.comp_hasStrictFDerivAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
    (hh : HasStrictDerivAt h₂ h₂' y) (hf : HasStrictFDerivAt f f' x) (hy : y = f x) :
    HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' y : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) (hh : HasStrictDerivAt h₂ h₂' y) (hf : HasStrictFDerivAt f f' x) (hy : y = f x)
Goal: HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x
-- HasDerivAt.comp_hasFDerivAt
theorem HasDerivAt.comp_hasFDerivAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
    (hh : HasDerivAt h₂ h₂' (f x)) (hf : HasFDerivAt f f' x) : HasFDerivAt (h₂ ∘ f) (h₂' • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) (hh : HasDerivAt h₂ h₂' (f x)) (hf : HasFDerivAt f f' x)
Goal: HasFDerivAt (h₂ ∘ f) (h₂' • f') x
-- HasDerivAt.comp_hasFDerivAt_of_eq
theorem HasDerivAt.comp_hasFDerivAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
    (hh : HasDerivAt h₂ h₂' y) (hf : HasFDerivAt f f' x) (hy : y = f x) :
    HasFDerivAt (h₂ ∘ f) (h₂' • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' y : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x : E) (hh : HasDerivAt h₂ h₂' y) (hf : HasFDerivAt f f' x) (hy : y = f x)
Goal: HasFDerivAt (h₂ ∘ f) (h₂' • f') x
-- HasDerivAt.comp_hasFDerivWithinAt
theorem HasDerivAt.comp_hasFDerivWithinAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s} (x)
    (hh : HasDerivAt h₂ h₂' (f x)) (hf : HasFDerivWithinAt f f' s x) :
    HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s : Set E} (x : E) (hh : HasDerivAt h₂ h₂' (f x)) (hf : HasFDerivWithinAt f f' s x)
Goal: HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x
-- HasDerivAt.comp_hasFDerivWithinAt_of_eq
theorem HasDerivAt.comp_hasFDerivWithinAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s} (x)
    (hh : HasDerivAt h₂ h₂' y) (hf : HasFDerivWithinAt f f' s x) (hy : y = f x) :
    HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' y : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s : Set E} (x : E) (hh : HasDerivAt h₂ h₂' y) (hf : HasFDerivWithinAt f f' s x) (hy : y = f x)
Goal: HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x
-- HasDerivWithinAt.comp_hasFDerivWithinAt
theorem HasDerivWithinAt.comp_hasFDerivWithinAt {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s t} (x)
    (hh : HasDerivWithinAt h₂ h₂' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
    HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s : Set E} {t : Set 𝕜'} (x : E) (hh : HasDerivWithinAt h₂ h₂' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : Set.MapsTo f s t)
Goal: HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x
-- HasDerivWithinAt.comp_hasFDerivWithinAt_of_eq
theorem HasDerivWithinAt.comp_hasFDerivWithinAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s t} (x)
    (hh : HasDerivWithinAt h₂ h₂' t y) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t)
    (hy : y = f x) :
    HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h₂ : 𝕜' → 𝕜'} {h₂' y : 𝕜'} {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s : Set E} {t : Set 𝕜'} (x : E) (hh : HasDerivWithinAt h₂ h₂' t y) (hf : HasFDerivWithinAt f f' s x) (hst : Set.MapsTo f s t) (hy : y = f x)
Goal: HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x
-- HasDerivAtFilter.comp
theorem HasDerivAtFilter.comp (hh₂ : HasDerivAtFilter h₂ h₂' (h x) L')
    (hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
    HasDerivAtFilter (h₂ ∘ h) (h₂' * h') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {L : Filter 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} {L' : Filter 𝕜'} (hh₂ : HasDerivAtFilter h₂ h₂' (h x) L') (hh : HasDerivAtFilter h h' x L) (hL : Filter.Tendsto h L L')
Goal: HasDerivAtFilter (h₂ ∘ h) (h₂' * h') x L
-- HasDerivAtFilter.comp_of_eq
theorem HasDerivAtFilter.comp_of_eq (hh₂ : HasDerivAtFilter h₂ h₂' y L')
    (hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') (hy : y = h x) :
    HasDerivAtFilter (h₂ ∘ h) (h₂' * h') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {L : Filter 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} {L' : Filter 𝕜'} {y : 𝕜'} (hh₂ : HasDerivAtFilter h₂ h₂' y L') (hh : HasDerivAtFilter h h' x L) (hL : Filter.Tendsto h L L') (hy : y = h x)
Goal: HasDerivAtFilter (h₂ ∘ h) (h₂' * h') x L
-- HasDerivWithinAt.comp
theorem HasDerivWithinAt.comp (hh₂ : HasDerivWithinAt h₂ h₂' s' (h x))
    (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s s') :
    HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} (hh₂ : HasDerivWithinAt h₂ h₂' s' (h x)) (hh : HasDerivWithinAt h h' s x) (hst : Set.MapsTo h s s')
Goal: HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x
-- HasDerivWithinAt.comp_of_eq
theorem HasDerivWithinAt.comp_of_eq (hh₂ : HasDerivWithinAt h₂ h₂' s' y)
    (hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s s') (hy : y = h x) :
    HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {s : Set 𝕜} {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {s' : Set 𝕜'} {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' y : 𝕜'} (hh₂ : HasDerivWithinAt h₂ h₂' s' y) (hh : HasDerivWithinAt h h' s x) (hst : Set.MapsTo h s s') (hy : y = h x)
Goal: HasDerivWithinAt (h₂ ∘ h) (h₂' * h') s x
-- HasDerivAt.comp
nonrec theorem HasDerivAt.comp (hh₂ : HasDerivAt h₂ h₂' (h x)) (hh : HasDerivAt h h' x) :
    HasDerivAt (h₂ ∘ h) (h₂' * h') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} (hh₂ : HasDerivAt h₂ h₂' (h x)) (hh : HasDerivAt h h' x)
Goal: HasDerivAt (h₂ ∘ h) (h₂' * h') x
-- HasDerivAt.comp_of_eq
theorem HasDerivAt.comp_of_eq
    (hh₂ : HasDerivAt h₂ h₂' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
    HasDerivAt (h₂ ∘ h) (h₂' * h') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' y : 𝕜'} (hh₂ : HasDerivAt h₂ h₂' y) (hh : HasDerivAt h h' x) (hy : y = h x)
Goal: HasDerivAt (h₂ ∘ h) (h₂' * h') x
-- HasStrictDerivAt.comp
theorem HasStrictDerivAt.comp (hh₂ : HasStrictDerivAt h₂ h₂' (h x)) (hh : HasStrictDerivAt h h' x) :
    HasStrictDerivAt (h₂ ∘ h) (h₂' * h') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] (x : 𝕜) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {h : 𝕜 → 𝕜'} {h₂ : 𝕜' → 𝕜'} {h' h₂' : 𝕜'} (hh₂ : HasStrictDerivAt h₂ h₂' (h x)) (hh : HasStrictDerivAt h h' x)
Goal: HasStrictDerivAt (h₂ ∘ h) (h₂' * h') x