ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1
lemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux1 [SigmaFinite μ]
    {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) (μ.prod volume))
    (hfg' : Integrable (fun x ↦ B (f x) (g' x)) (μ.prod volume))
    (hfg : Integrable (fun x ↦ B (f x) (g x)) (μ.prod volume))
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x (0, 1)) :
    ∫ x, B (f x) (g' x) ∂(μ.prod volume) = - ∫ x, B (f' x) (g x) ∂(μ.prod volume) 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] {μ : MeasureTheory.Measure E} [MeasureTheory.SigmaFinite μ] {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) (μ.prod MeasureTheory.volume)) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) (μ.prod MeasureTheory.volume)) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) (μ.prod MeasureTheory.volume)) (hf : ∀ (x : E × ℝ), HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ (x : E × ℝ), HasLineDerivAt ℝ g (g' x) x (0, 1))
Goal: ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ.prod MeasureTheory.volume =
  -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ.prod MeasureTheory.volume
-- integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
lemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
    [FiniteDimensional ℝ E] {μ : Measure (E × ℝ)} [IsAddHaarMeasure μ]
    {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x (0, 1)) :
    ∫ x, B (f x) (g' x) ∂μ = - ∫ x, B (f' x) (g x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure (E × ℝ)} [μ.IsAddHaarMeasure] {f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E × ℝ), HasLineDerivAt ℝ f (f' x) x (0, 1)) (hg : ∀ (x : E × ℝ), HasLineDerivAt ℝ g (g' x) x (0, 1))
Goal: ∫ (x : E × ℝ), (B (f x)) (g' x) ∂μ = -∫ (x : E × ℝ), (B (f' x)) (g x) ∂μ
-- integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
    {f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasLineDerivAt ℝ f (f' x) x v) (hg : ∀ x, HasLineDerivAt ℝ g (g' x) x v) :
    ∫ x, B (f x) (g' x) ∂μ = - ∫ x, B (f' x) (g x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B (f' x)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) (g' x)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E), HasLineDerivAt ℝ f (f' x) x v) (hg : ∀ (x : E), HasLineDerivAt ℝ g (g' x) x v)
Goal: ∫ (x : E), (B (f x)) (g' x) ∂μ = -∫ (x : E), (B (f' x)) (g x) ∂μ
-- integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable
theorem integral_bilinear_hasFDerivAt_right_eq_neg_left_of_integrable
    {f : E → F} {f' : E → (E →L[ℝ] F)}
    {g : E → G} {g' : E → (E →L[ℝ] G)} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (f' x v) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (g' x v)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : ∀ x, HasFDerivAt f (f' x) x) (hg : ∀ x, HasFDerivAt g (g' x) x) :
    ∫ x, B (f x) (g' x v) ∂μ = - ∫ x, B (f' x v) (g x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {f : E → F} {f' : E → E →L[ℝ] F} {g : E → G} {g' : E → E →L[ℝ] G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B ((f' x) v)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) ((g' x) v)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : ∀ (x : E), HasFDerivAt f (f' x) x) (hg : ∀ (x : E), HasFDerivAt g (g' x) x)
Goal: ∫ (x : E), (B (f x)) ((g' x) v) ∂μ = -∫ (x : E), (B ((f' x) v)) (g x) ∂μ
-- integral_bilinear_fderiv_right_eq_neg_left_of_integrable
theorem integral_bilinear_fderiv_right_eq_neg_left_of_integrable
    {f : E → F} {g : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
    (hf'g : Integrable (fun x ↦ B (fderiv ℝ f x v) (g x)) μ)
    (hfg' : Integrable (fun x ↦ B (f x) (fderiv ℝ g x v)) μ)
    (hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, B (f x) (fderiv ℝ g x v) ∂μ = - ∫ x, B (fderiv ℝ f x v) (g x) ∂μ 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} {W : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {f : E → F} {g : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W} (hf'g : MeasureTheory.Integrable (fun x => (B ((fderiv ℝ f x) v)) (g x)) μ) (hfg' : MeasureTheory.Integrable (fun x => (B (f x)) ((fderiv ℝ g x) v)) μ) (hfg : MeasureTheory.Integrable (fun x => (B (f x)) (g x)) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
Goal: ∫ (x : E), (B (f x)) ((fderiv ℝ g x) v) ∂μ = -∫ (x : E), (B ((fderiv ℝ f x) v)) (g x) ∂μ
-- integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable
theorem integral_smul_fderiv_eq_neg_fderiv_smul_of_integrable
    {f : E → 𝕜} {g : E → G} {v : E}
    (hf'g : Integrable (fun x ↦ fderiv ℝ f x v • g x) μ)
    (hfg' : Integrable (fun x ↦ f x • fderiv ℝ g x v) μ)
    (hfg : Integrable (fun x ↦ f x • g x) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, f x • fderiv ℝ g x v ∂μ = - ∫ x, fderiv ℝ f x v • g x ∂μ 
Arguments: {E : Type u_1} {G : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup G] [NormedSpace ℝ G] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {𝕜 : Type u_5} [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] [NormedSpace 𝕜 G] [IsScalarTower ℝ 𝕜 G] {f : E → 𝕜} {g : E → G} {v : E} (hf'g : MeasureTheory.Integrable (fun x => (fderiv ℝ f x) v • g x) μ) (hfg' : MeasureTheory.Integrable (fun x => f x • (fderiv ℝ g x) v) μ) (hfg : MeasureTheory.Integrable (fun x => f x • g x) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
Goal: ∫ (x : E), f x • (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v • g x ∂μ
-- integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
theorem integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
    {f : E → 𝕜} {g : E → 𝕜} {v : E}
    (hf'g : Integrable (fun x ↦ fderiv ℝ f x v * g x) μ)
    (hfg' : Integrable (fun x ↦ f x * fderiv ℝ g x v) μ)
    (hfg : Integrable (fun x ↦ f x * g x) μ)
    (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) :
    ∫ x, f x * fderiv ℝ g x v ∂μ = - ∫ x, fderiv ℝ f x v * g x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {𝕜 : Type u_5} [NormedField 𝕜] [NormedAlgebra ℝ 𝕜] {f g : E → 𝕜} {v : E} (hf'g : MeasureTheory.Integrable (fun x => (fderiv ℝ f x) v * g x) μ) (hfg' : MeasureTheory.Integrable (fun x => f x * (fderiv ℝ g x) v) μ) (hfg : MeasureTheory.Integrable (fun x => f x * g x) μ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
Goal: ∫ (x : E), f x * (fderiv ℝ g x) v ∂μ = -∫ (x : E), (fderiv ℝ f x) v * g x ∂μ
-- ContDiffAt.toPartialHomeomorph_coe
@[simp]
theorem toPartialHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    (hf.toPartialHomeomorph f hf' hn : E → F) = f 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕂 F] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕂] F} {a : E} {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : 1 ≤ n)
Goal: ↑(ContDiffAt.toPartialHomeomorph f hf hf' hn) = f
-- ContDiffAt.mem_toPartialHomeomorph_source
theorem mem_toPartialHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    a ∈ (hf.toPartialHomeomorph f hf' hn).source 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕂 F] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕂] F} {a : E} {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : 1 ≤ n)
Goal: a ∈ (ContDiffAt.toPartialHomeomorph f hf hf' hn).source
-- ContDiffAt.image_mem_toPartialHomeomorph_target
theorem image_mem_toPartialHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    f a ∈ (hf.toPartialHomeomorph f hf' hn).target 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕂 F] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕂] F} {a : E} {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : 1 ≤ n)
Goal: f a ∈ (ContDiffAt.toPartialHomeomorph f hf hf' hn).target
-- ContDiffAt.localInverse_apply_image
theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕂 F] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕂] F} {a : E} {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : 1 ≤ n)
Goal: hf.localInverse hf' hn (f a) = a
-- ContDiffAt.to_localInverse
theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
    (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
    ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) 
Arguments: {𝕂 : Type u_1} [RCLike 𝕂] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕂 F] [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕂] F} {a : E} {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : 1 ≤ n)
Goal: ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a)
-- tendsto_apply_add_mul_sq_div_sub
theorem tendsto_apply_add_mul_sq_div_sub {f : ℝ → ℝ} {x a c d : ℝ} {l : Filter ℝ} (hl : l ≤ 𝓝[≠] x)
    (hf : Tendsto (fun y => (f y - d) / (y - x)) l (𝓝 a))
    (h' : Tendsto (fun y => y + c * (y - x) ^ 2) l l) :
    Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a) 
Arguments: {f : ℝ → ℝ} {x a c d : ℝ} {l : Filter ℝ} (hl : l ≤ 𝓝[≠] x) (hf : Filter.Tendsto (fun y => (f y - d) / (y - x)) l (𝓝 a)) (h' : Filter.Tendsto (fun y => y + c * (y - x) ^ 2) l l)
Goal: Filter.Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a)
-- StieltjesFunction.ae_hasDerivAt
theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
    ∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x 
Arguments: (f : StieltjesFunction)
Goal: ∀ᵐ (x : ℝ), HasDerivAt (↑f) (f.measure.rnDeriv MeasureTheory.volume x).toReal x
-- Monotone.ae_hasDerivAt
theorem Monotone.ae_hasDerivAt {f : ℝ → ℝ} (hf : Monotone f) :
    ∀ᵐ x, HasDerivAt f (rnDeriv hf.stieltjesFunction.measure volume x).toReal x 
Arguments: {f : ℝ → ℝ} (hf : Monotone f)
Goal: ∀ᵐ (x : ℝ), HasDerivAt f (hf.stieltjesFunction.measure.rnDeriv MeasureTheory.volume x).toReal x
-- Monotone.ae_differentiableAt
theorem Monotone.ae_differentiableAt {f : ℝ → ℝ} (hf : Monotone f) :
    ∀ᵐ x, DifferentiableAt ℝ f x 
Arguments: {f : ℝ → ℝ} (hf : Monotone f)
Goal: ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x
-- MonotoneOn.ae_differentiableWithinAt_of_mem
theorem MonotoneOn.ae_differentiableWithinAt_of_mem {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s) :
    ∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s)
Goal: ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x
-- MonotoneOn.ae_differentiableWithinAt
theorem MonotoneOn.ae_differentiableWithinAt {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s)
    (hs : MeasurableSet s) : ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x 
Arguments: {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s) (hs : MeasurableSet s)
Goal: ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x
-- ContinuousLinearMap.hasDerivWithinAt_of_bilinear
theorem hasDerivWithinAt_of_bilinear
    (hu : HasDerivWithinAt u u' s x) (hv : HasDerivWithinAt v v' s x) :
    HasDerivWithinAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : 𝕜} {s : Set 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E} {v : 𝕜 → F} {u' : E} {v' : F} (hu : HasDerivWithinAt u u' s x) (hv : HasDerivWithinAt v v' s x)
Goal: HasDerivWithinAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) s x
-- ContinuousLinearMap.hasDerivAt_of_bilinear
theorem hasDerivAt_of_bilinear (hu : HasDerivAt u u' x) (hv : HasDerivAt v v' x) :
    HasDerivAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E} {v : 𝕜 → F} {u' : E} {v' : F} (hu : HasDerivAt u u' x) (hv : HasDerivAt v v' x)
Goal: HasDerivAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x
-- ContinuousLinearMap.hasStrictDerivAt_of_bilinear
theorem hasStrictDerivAt_of_bilinear (hu : HasStrictDerivAt u u' x) (hv : HasStrictDerivAt v v' x) :
    HasStrictDerivAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E} {v : 𝕜 → F} {u' : E} {v' : F} (hu : HasStrictDerivAt u u' x) (hv : HasStrictDerivAt v v' x)
Goal: HasStrictDerivAt (fun x => (B (u x)) (v x)) ((B (u x)) v' + (B u') (v x)) x
-- ContinuousLinearMap.derivWithin_of_bilinear
theorem derivWithin_of_bilinear (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hu : DifferentiableWithinAt 𝕜 u s x) (hv : DifferentiableWithinAt 𝕜 v s x) :
    derivWithin (fun y => B (u y) (v y)) s x =
      B (u x) (derivWithin v s x) + B (derivWithin u s x) (v x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : 𝕜} {s : Set 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E} {v : 𝕜 → F} (hxs : UniqueDiffWithinAt 𝕜 s x) (hu : DifferentiableWithinAt 𝕜 u s x) (hv : DifferentiableWithinAt 𝕜 v s x)
Goal: derivWithin (fun y => (B (u y)) (v y)) s x = (B (u x)) (derivWithin v s x) + (B (derivWithin u s x)) (v x)
-- ContinuousLinearMap.deriv_of_bilinear
theorem deriv_of_bilinear (hu : DifferentiableAt 𝕜 u x) (hv : DifferentiableAt 𝕜 v x) :
    deriv (fun y => B (u y) (v y)) x = B (u x) (deriv v x) + B (deriv u x) (v x) 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {E : Type w} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {G : Type u_1} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {x : 𝕜} {B : E →L[𝕜] F →L[𝕜] G} {u : 𝕜 → E} {v : 𝕜 → F} (hu : DifferentiableAt 𝕜 u x) (hv : DifferentiableAt 𝕜 v x)
Goal: deriv (fun y => (B (u y)) (v y)) x = (B (u x)) (deriv v x) + (B (deriv u x)) (v x)
-- HasDerivWithinAt.smul
theorem HasDerivWithinAt.smul (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun y => c y • f y) (c x • f' + c' • f x) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun y => c y • f y) (c x • f' + c' • f x) s x
-- HasDerivAt.smul
theorem HasDerivAt.smul (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x) :
    HasDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x)
Goal: HasDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x
-- HasStrictDerivAt.smul
nonrec theorem HasStrictDerivAt.smul (hc : HasStrictDerivAt c c' x) (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasStrictDerivAt c c' x) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x
-- derivWithin_smul
theorem derivWithin_smul (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x)
    (hf : DifferentiableWithinAt 𝕜 f s x) :
    derivWithin (fun y => c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (hf : DifferentiableWithinAt 𝕜 f s x)
Goal: derivWithin (fun y => c y • f y) s x = c x • derivWithin f s x + derivWithin c s x • f x
-- deriv_smul
theorem deriv_smul (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x) :
    deriv (fun y => c y • f y) x = c x • deriv f x + deriv c x • f x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (hf : DifferentiableAt 𝕜 f x)
Goal: deriv (fun y => c y • f y) x = c x • deriv f x + deriv c x • f x
-- HasStrictDerivAt.smul_const
theorem HasStrictDerivAt.smul_const (hc : HasStrictDerivAt c c' x) (f : F) :
    HasStrictDerivAt (fun y => c y • f) (c' • f) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasStrictDerivAt c c' x) (f : F)
Goal: HasStrictDerivAt (fun y => c y • f) (c' • f) x
-- HasDerivWithinAt.smul_const
theorem HasDerivWithinAt.smul_const (hc : HasDerivWithinAt c c' s x) (f : F) :
    HasDerivWithinAt (fun y => c y • f) (c' • f) s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivWithinAt c c' s x) (f : F)
Goal: HasDerivWithinAt (fun y => c y • f) (c' • f) s x
-- HasDerivAt.smul_const
theorem HasDerivAt.smul_const (hc : HasDerivAt c c' x) (f : F) :
    HasDerivAt (fun y => c y • f) (c' • f) x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} {c' : 𝕜'} (hc : HasDerivAt c c' x) (f : F)
Goal: HasDerivAt (fun y => c y • f) (c' • f) x
-- derivWithin_smul_const
theorem derivWithin_smul_const (hxs : UniqueDiffWithinAt 𝕜 s x)
    (hc : DifferentiableWithinAt 𝕜 c s x) (f : F) :
    derivWithin (fun y => c y • f) s x = derivWithin c s x • f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {s : Set 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} (hxs : UniqueDiffWithinAt 𝕜 s x) (hc : DifferentiableWithinAt 𝕜 c s x) (f : F)
Goal: derivWithin (fun y => c y • f) s x = derivWithin c s x • f
-- deriv_smul_const
theorem deriv_smul_const (hc : DifferentiableAt 𝕜 c x) (f : F) :
    deriv (fun y => c y • f) x = deriv c x • f 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : 𝕜} {𝕜' : Type u_2} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F] {c : 𝕜 → 𝕜'} (hc : DifferentiableAt 𝕜 c x) (f : F)
Goal: deriv (fun y => c y • f) x = deriv c x • f
-- HasStrictDerivAt.const_smul
nonrec theorem HasStrictDerivAt.const_smul (c : R) (hf : HasStrictDerivAt f f' x) :
    HasStrictDerivAt (fun y => c • f y) (c • f') x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R) (hf : HasStrictDerivAt f f' x)
Goal: HasStrictDerivAt (fun y => c • f y) (c • f') x
-- HasDerivAtFilter.const_smul
nonrec theorem HasDerivAtFilter.const_smul (c : R) (hf : HasDerivAtFilter f f' x L) :
    HasDerivAtFilter (fun y => c • f y) (c • f') x L 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {L : Filter 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R) (hf : HasDerivAtFilter f f' x L)
Goal: HasDerivAtFilter (fun y => c • f y) (c • f') x L
-- HasDerivWithinAt.const_smul
nonrec theorem HasDerivWithinAt.const_smul (c : R) (hf : HasDerivWithinAt f f' s x) :
    HasDerivWithinAt (fun y => c • f y) (c • f') s x 
Arguments: {𝕜 : Type u} [NontriviallyNormedField 𝕜] {F : Type v} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} {x : 𝕜} {s : Set 𝕜} {R : Type u_2} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F] (c : R) (hf : HasDerivWithinAt f f' s x)
Goal: HasDerivWithinAt (fun y => c • f y) (c • f') s x