ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integral_eq_zero_of_hasDerivAt_of_integrable
theorem integral_eq_zero_of_hasDerivAt_of_integrable
    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f') (hf : Integrable f) :
    ∫ x, f' x = 0 
Arguments: {E : Type u_1} {f f' : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] (hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : MeasureTheory.Integrable f' MeasureTheory.volume) (hf : MeasureTheory.Integrable f MeasureTheory.volume)
Goal: ∫ (x : ℝ), f' x = 0
-- MeasureTheory.integral_comp_smul_deriv_Ioi
theorem integral_comp_smul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → E} {a : ℝ}
    (hf : ContinuousOn f <| Ici a) (hft : Tendsto f atTop atTop)
    (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
    (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) (Ici a)) :
    (∫ x in Ioi a, f' x • (g ∘ f) x) = ∫ u in Ioi (f a), g u 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → ℝ} {g : ℝ → E} {a : ℝ} (hf : ContinuousOn f (Set.Ici a)) (hft : Filter.Tendsto f Filter.atTop Filter.atTop) (hff' : ∀ x ∈ Set.Ioi a, HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hg_cont : ContinuousOn g (f '' Set.Ioi a)) (hg1 : MeasureTheory.IntegrableOn g (f '' Set.Ici a) MeasureTheory.volume) (hg2 : MeasureTheory.IntegrableOn (fun x => f' x • (g ∘ f) x) (Set.Ici a) MeasureTheory.volume)
Goal: ∫ (x : ℝ) in Set.Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Set.Ioi (f a), g u
-- MeasureTheory.integral_comp_mul_deriv_Ioi
theorem integral_comp_mul_deriv_Ioi {f f' : ℝ → ℝ} {g : ℝ → ℝ} {a : ℝ}
    (hf : ContinuousOn f <| Ici a) (hft : Tendsto f atTop atTop)
    (hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g <| f '' Ioi a) (hg1 : IntegrableOn g <| f '' Ici a)
    (hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) (Ici a)) :
    (∫ x in Ioi a, (g ∘ f) x * f' x) = ∫ u in Ioi (f a), g u 
Arguments: {f f' g : ℝ → ℝ} {a : ℝ} (hf : ContinuousOn f (Set.Ici a)) (hft : Filter.Tendsto f Filter.atTop Filter.atTop) (hff' : ∀ x ∈ Set.Ioi a, HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hg_cont : ContinuousOn g (f '' Set.Ioi a)) (hg1 : MeasureTheory.IntegrableOn g (f '' Set.Ici a) MeasureTheory.volume) (hg2 : MeasureTheory.IntegrableOn (fun x => (g ∘ f) x * f' x) (Set.Ici a) MeasureTheory.volume)
Goal: ∫ (x : ℝ) in Set.Ioi a, (g ∘ f) x * f' x = ∫ (u : ℝ) in Set.Ioi (f a), g u
-- MeasureTheory.integral_comp_rpow_Ioi
theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    (∫ x in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (g : ℝ → E) {p : ℝ} (hp : p ≠ 0)
Goal: ∫ (x : ℝ) in Set.Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Set.Ioi 0, g y
-- MeasureTheory.integral_comp_rpow_Ioi_of_pos
theorem integral_comp_rpow_Ioi_of_pos {g : ℝ → E} {p : ℝ} (hp : 0 < p) :
    (∫ x in Ioi 0, (p * x ^ (p - 1)) • g (x ^ p)) = ∫ y in Ioi 0, g y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {g : ℝ → E} {p : ℝ} (hp : 0 < p)
Goal: ∫ (x : ℝ) in Set.Ioi 0, (p * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Set.Ioi 0, g y
-- MeasureTheory.integral_comp_mul_left_Ioi
theorem integral_comp_mul_left_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
    (∫ x in Ioi a, g (b * x)) = b⁻¹ • ∫ x in Ioi (b * a), g x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b)
Goal: ∫ (x : ℝ) in Set.Ioi a, g (b * x) = b⁻¹ • ∫ (x : ℝ) in Set.Ioi (b * a), g x
-- MeasureTheory.integral_comp_mul_right_Ioi
theorem integral_comp_mul_right_Ioi (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b) :
    (∫ x in Ioi a, g (x * b)) = b⁻¹ • ∫ x in Ioi (a * b), g x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (g : ℝ → E) (a : ℝ) {b : ℝ} (hb : 0 < b)
Goal: ∫ (x : ℝ) in Set.Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Set.Ioi (a * b), g x
-- MeasureTheory.integrableOn_Ioi_comp_rpow_iff
theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0)
Goal: MeasureTheory.IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Set.Ioi 0) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi 0) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioi_comp_rpow_iff'
theorem integrableOn_Ioi_comp_rpow_iff' [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
    IntegrableOn (fun x => x ^ (p - 1) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0)
Goal: MeasureTheory.IntegrableOn (fun x => x ^ (p - 1) • f (x ^ p)) (Set.Ioi 0) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi 0) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioi_comp_mul_left_iff
theorem integrableOn_Ioi_comp_mul_left_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
    IntegrableOn (fun x => f (a * x)) (Ioi c) ↔ IntegrableOn f (Ioi <| a * c) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a)
Goal: MeasureTheory.IntegrableOn (fun x => f (a * x)) (Set.Ioi c) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi (a * c)) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioi_comp_mul_right_iff
theorem integrableOn_Ioi_comp_mul_right_iff (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a) :
    IntegrableOn (fun x => f (x * a)) (Ioi c) ↔ IntegrableOn f (Ioi <| c * a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℝ → E) (c : ℝ) {a : ℝ} (ha : 0 < a)
Goal: MeasureTheory.IntegrableOn (fun x => f (x * a)) (Set.Ioi c) MeasureTheory.volume ↔
  MeasureTheory.IntegrableOn f (Set.Ioi (c * a)) MeasureTheory.volume
-- MeasureTheory.integral_bilinear_hasDerivAt_eq_sub
theorem integral_bilinear_hasDerivAt_eq_sub [CompleteSpace G]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv : Integrable (fun x ↦ L (u x) (v' x) + L (u' x) (v x)))
    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
    ∫ (x : ℝ), L (u x) (v' x) + L (u' x) (v x) = n - m 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] {L : E →L[ℝ] F →L[ℝ] G} {u : ℝ → E} {v : ℝ → F} {u' : ℝ → E} {v' : ℝ → F} {m n : G} [CompleteSpace G] (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv : MeasureTheory.Integrable (fun x => (L (u x)) (v' x) + (L (u' x)) (v x)) MeasureTheory.volume) (h_bot : Filter.Tendsto (fun x => (L (u x)) (v x)) Filter.atBot (𝓝 m)) (h_top : Filter.Tendsto (fun x => (L (u x)) (v x)) Filter.atTop (𝓝 n))
Goal: ∫ (x : ℝ), (L (u x)) (v' x) + (L (u' x)) (v x) = n - m
-- MeasureTheory.integral_bilinear_hasDerivAt_right_eq_sub
theorem integral_bilinear_hasDerivAt_right_eq_sub [CompleteSpace G]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
    (h_bot : Tendsto (fun x ↦ L (u x) (v x)) atBot (𝓝 m))
    (h_top : Tendsto (fun x ↦ L (u x) (v x)) atTop (𝓝 n)) :
    ∫ (x : ℝ), L (u x) (v' x) = n - m - ∫ (x : ℝ), L (u' x) (v x) 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] {L : E →L[ℝ] F →L[ℝ] G} {u : ℝ → E} {v : ℝ → F} {u' : ℝ → E} {v' : ℝ → F} {m n : G} [CompleteSpace G] (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv' : MeasureTheory.Integrable (fun x => (L (u x)) (v' x)) MeasureTheory.volume) (hu'v : MeasureTheory.Integrable (fun x => (L (u' x)) (v x)) MeasureTheory.volume) (h_bot : Filter.Tendsto (fun x => (L (u x)) (v x)) Filter.atBot (𝓝 m)) (h_top : Filter.Tendsto (fun x => (L (u x)) (v x)) Filter.atTop (𝓝 n))
Goal: ∫ (x : ℝ), (L (u x)) (v' x) = n - m - ∫ (x : ℝ), (L (u' x)) (v x)
-- MeasureTheory.integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
theorem integral_bilinear_hasDerivAt_right_eq_neg_left_of_integrable
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (fun x ↦ L (u x) (v' x))) (hu'v : Integrable (fun x ↦ L (u' x) (v x)))
    (huv : Integrable (fun x ↦ L (u x) (v x))) :
    ∫ (x : ℝ), L (u x) (v' x) = - ∫ (x : ℝ), L (u' x) (v x) 
Arguments: {E : Type u_1} {F : Type u_2} {G : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] {L : E →L[ℝ] F →L[ℝ] G} {u : ℝ → E} {v : ℝ → F} {u' : ℝ → E} {v' : ℝ → F} (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv' : MeasureTheory.Integrable (fun x => (L (u x)) (v' x)) MeasureTheory.volume) (hu'v : MeasureTheory.Integrable (fun x => (L (u' x)) (v x)) MeasureTheory.volume) (huv : MeasureTheory.Integrable (fun x => (L (u x)) (v x)) MeasureTheory.volume)
Goal: ∫ (x : ℝ), (L (u x)) (v' x) = -∫ (x : ℝ), (L (u' x)) (v x)
-- MeasureTheory.integral_deriv_mul_eq_sub
theorem integral_deriv_mul_eq_sub [CompleteSpace A]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv : Integrable (u' * v + u * v'))
    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a' 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv : MeasureTheory.Integrable (u' * v + u * v') MeasureTheory.volume) (h_bot : Filter.Tendsto (u * v) Filter.atBot (𝓝 a')) (h_top : Filter.Tendsto (u * v) Filter.atTop (𝓝 b'))
Goal: ∫ (x : ℝ), u' x * v x + u x * v' x = b' - a'
-- MeasureTheory.integral_mul_deriv_eq_deriv_mul
theorem integral_mul_deriv_eq_deriv_mul [CompleteSpace A]
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v))
    (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv' : MeasureTheory.Integrable (u * v') MeasureTheory.volume) (hu'v : MeasureTheory.Integrable (u' * v) MeasureTheory.volume) (h_bot : Filter.Tendsto (u * v) Filter.atBot (𝓝 a')) (h_top : Filter.Tendsto (u * v) Filter.atTop (𝓝 b'))
Goal: ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x
-- MeasureTheory.integral_mul_deriv_eq_deriv_mul_of_integrable
theorem integral_mul_deriv_eq_deriv_mul_of_integrable
    (hu : ∀ x, HasDerivAt u (u' x) x) (hv : ∀ x, HasDerivAt v (v' x) x)
    (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v)) (huv : Integrable (u * v)) :
    ∫ (x : ℝ), u x * v' x = - ∫ (x : ℝ), u' x * v x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {u v u' v' : ℝ → A} (hu : ∀ (x : ℝ), HasDerivAt u (u' x) x) (hv : ∀ (x : ℝ), HasDerivAt v (v' x) x) (huv' : MeasureTheory.Integrable (u * v') MeasureTheory.volume) (hu'v : MeasureTheory.Integrable (u' * v) MeasureTheory.volume) (huv : MeasureTheory.Integrable (u * v) MeasureTheory.volume)
Goal: ∫ (x : ℝ), u x * v' x = -∫ (x : ℝ), u' x * v x
-- MeasureTheory.integral_Ioi_deriv_mul_eq_sub
theorem integral_Ioi_deriv_mul_eq_sub
    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
    (huv : IntegrableOn (u' * v + u * v') (Ioi a))
    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a' 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a : ℝ} {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ x ∈ Set.Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Set.Ioi a, HasDerivAt v (v' x) x) (huv : MeasureTheory.IntegrableOn (u' * v + u * v') (Set.Ioi a) MeasureTheory.volume) (h_zero : Filter.Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Filter.Tendsto (u * v) Filter.atTop (𝓝 b'))
Goal: ∫ (x : ℝ) in Set.Ioi a, u' x * v x + u x * v' x = b' - a'
-- MeasureTheory.integral_Ioi_mul_deriv_eq_deriv_mul
theorem integral_Ioi_mul_deriv_eq_deriv_mul
    (hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
    (huv' : IntegrableOn (u * v') (Ioi a)) (hu'v : IntegrableOn (u' * v) (Ioi a))
    (h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
    ∫ (x : ℝ) in Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Ioi a, u' x * v x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a : ℝ} {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ x ∈ Set.Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Set.Ioi a, HasDerivAt v (v' x) x) (huv' : MeasureTheory.IntegrableOn (u * v') (Set.Ioi a) MeasureTheory.volume) (hu'v : MeasureTheory.IntegrableOn (u' * v) (Set.Ioi a) MeasureTheory.volume) (h_zero : Filter.Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Filter.Tendsto (u * v) Filter.atTop (𝓝 b'))
Goal: ∫ (x : ℝ) in Set.Ioi a, u x * v' x = b' - a' - ∫ (x : ℝ) in Set.Ioi a, u' x * v x
-- MeasureTheory.integral_Iic_deriv_mul_eq_sub
theorem integral_Iic_deriv_mul_eq_sub
    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
    (huv : IntegrableOn (u' * v + u * v') (Iic a))
    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
    ∫ (x : ℝ) in Iic a, u' x * v x + u x * v' x = a' - b' 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a : ℝ} {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ x ∈ Set.Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Set.Iio a, HasDerivAt v (v' x) x) (huv : MeasureTheory.IntegrableOn (u' * v + u * v') (Set.Iic a) MeasureTheory.volume) (h_zero : Filter.Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Filter.Tendsto (u * v) Filter.atBot (𝓝 b'))
Goal: ∫ (x : ℝ) in Set.Iic a, u' x * v x + u x * v' x = a' - b'
-- MeasureTheory.integral_Iic_mul_deriv_eq_deriv_mul
theorem integral_Iic_mul_deriv_eq_deriv_mul
    (hu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x)
    (huv' : IntegrableOn (u * v') (Iic a)) (hu'v : IntegrableOn (u' * v) (Iic a))
    (h_zero : Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Tendsto (u * v) atBot (𝓝 b')) :
    ∫ (x : ℝ) in Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Iic a, u' x * v x 
Arguments: {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] {a : ℝ} {a' b' : A} {u v u' v' : ℝ → A} [CompleteSpace A] (hu : ∀ x ∈ Set.Iio a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Set.Iio a, HasDerivAt v (v' x) x) (huv' : MeasureTheory.IntegrableOn (u * v') (Set.Iic a) MeasureTheory.volume) (hu'v : MeasureTheory.IntegrableOn (u' * v) (Set.Iic a) MeasureTheory.volume) (h_zero : Filter.Tendsto (u * v) (𝓝[<] a) (𝓝 a')) (h_infty : Filter.Tendsto (u * v) Filter.atBot (𝓝 b'))
Goal: ∫ (x : ℝ) in Set.Iic a, u x * v' x = a' - b' - ∫ (x : ℝ) in Set.Iic a, u' x * v x
-- MeasureTheory.laverage_zero
@[simp]
theorem laverage_zero : ⨍⁻ _x, (0 : ℝ≥0∞) ∂μ = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α)
Goal: ⨍⁻ (_x : α), 0 ∂μ = 0
-- MeasureTheory.laverage_zero_measure
@[simp]
theorem laverage_zero_measure (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂(0 : Measure α) = 0 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (f : α → ℝ≥0∞)
Goal: ⨍⁻ (x : α), f x ∂0 = 0
-- MeasureTheory.laverage_eq'
theorem laverage_eq' (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂(μ univ)⁻¹ • μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂(μ Set.univ)⁻¹ • μ
-- MeasureTheory.laverage_eq
theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞)
Goal: ⨍⁻ (x : α), f x ∂μ = (∫⁻ (x : α), f x ∂μ) / μ Set.univ
-- MeasureTheory.laverage_eq_lintegral
theorem laverage_eq_lintegral [IsProbabilityMeasure μ] (f : α → ℝ≥0∞) :
    ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsProbabilityMeasure μ] (f : α → ℝ≥0∞)
Goal: ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.measure_mul_laverage
@[simp]
theorem measure_mul_laverage [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
    μ univ * ⨍⁻ x, f x ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ℝ≥0∞)
Goal: μ Set.univ * ⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLaverage_eq
theorem setLaverage_eq (f : α → ℝ≥0∞) (s : Set α) :
    ⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞) (s : Set α)
Goal: ⨍⁻ (x : α) in s, f x ∂μ = (∫⁻ (x : α) in s, f x ∂μ) / μ s
-- MeasureTheory.setLaverage_eq'
theorem setLaverage_eq' (f : α → ℝ≥0∞) (s : Set α) :
    ⨍⁻ x in s, f x ∂μ = ∫⁻ x, f x ∂(μ s)⁻¹ • μ.restrict s 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) (f : α → ℝ≥0∞) (s : Set α)
Goal: ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α), f x ∂(μ s)⁻¹ • μ.restrict s
-- MeasureTheory.laverage_congr
theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g)
Goal: ⨍⁻ (x : α), f x ∂μ = ⨍⁻ (x : α), g x ∂μ
-- MeasureTheory.setLaverage_congr
theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → ℝ≥0∞} (h : s =ᵐ[μ] t)
Goal: ⨍⁻ (x : α) in s, f x ∂μ = ⨍⁻ (x : α) in t, f x ∂μ
-- MeasureTheory.setLaverage_congr_fun
theorem setLaverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
    ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in s, g x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f g : α → ℝ≥0∞} (hs : MeasurableSet s) (h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x)
Goal: ⨍⁻ (x : α) in s, f x ∂μ = ⨍⁻ (x : α) in s, g x ∂μ
-- MeasureTheory.laverage_lt_top
theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} (hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
Goal: ⨍⁻ (x : α), f x ∂μ < ⊤
-- MeasureTheory.setLaverage_lt_top
theorem setLaverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞}
Goal: ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤ → ⨍⁻ (x : α) in s, f x ∂μ < ⊤
-- MeasureTheory.laverage_add_measure
theorem laverage_add_measure :
    ⨍⁻ x, f x ∂(μ + ν) =
      μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ≥0∞}
Goal: ⨍⁻ (x : α), f x ∂(μ + ν) =
  μ Set.univ / (μ Set.univ + ν Set.univ) * ⨍⁻ (x : α), f x ∂μ +
    ν Set.univ / (μ Set.univ + ν Set.univ) * ⨍⁻ (x : α), f x ∂ν
-- MeasureTheory.measure_mul_setLaverage
theorem measure_mul_setLaverage (f : α → ℝ≥0∞) (h : μ s ≠ ∞) :
    μ s * ⨍⁻ x in s, f x ∂μ = ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (f : α → ℝ≥0∞) (h : μ s ≠ ⊤)
Goal: μ s * ⨍⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.laverage_union
theorem laverage_union (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
    ⨍⁻ x in s ∪ t, f x ∂μ =
      μ s / (μ s + μ t) * ⨍⁻ x in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ x in t, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → ℝ≥0∞} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ)
Goal: ⨍⁻ (x : α) in s ∪ t, f x ∂μ = μ s / (μ s + μ t) * ⨍⁻ (x : α) in s, f x ∂μ + μ t / (μ s + μ t) * ⨍⁻ (x : α) in t, f x ∂μ
-- MeasureTheory.laverage_union_mem_openSegment
theorem laverage_union_mem_openSegment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
    (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
    ⨍⁻ x in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in t, f x ∂μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → ℝ≥0∞} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hs₀ : μ s ≠ 0) (ht₀ : μ t ≠ 0) (hsμ : μ s ≠ ⊤) (htμ : μ t ≠ ⊤)
Goal: ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in t, f x ∂μ)
-- MeasureTheory.laverage_union_mem_segment
theorem laverage_union_mem_segment (hd : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
    (hsμ : μ s ≠ ∞) (htμ : μ t ≠ ∞) :
    ⨍⁻ x in s ∪ t, f x ∂μ ∈ [⨍⁻ x in s, f x ∂μ -[ℝ≥0∞] ⨍⁻ x in t, f x ∂μ] 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α} {f : α → ℝ≥0∞} (hd : MeasureTheory.AEDisjoint μ s t) (ht : MeasureTheory.NullMeasurableSet t μ) (hsμ : μ s ≠ ⊤) (htμ : μ t ≠ ⊤)
Goal: ⨍⁻ (x : α) in s ∪ t, f x ∂μ ∈ [⨍⁻ (x : α) in s, f x ∂μ-[ℝ≥0∞]⨍⁻ (x : α) in t, f x ∂μ]
-- MeasureTheory.laverage_mem_openSegment_compl_self
theorem laverage_mem_openSegment_compl_self [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ)
    (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0) :
    ⨍⁻ x, f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ x in s, f x ∂μ) (⨍⁻ x in sᶜ, f x ∂μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} [MeasureTheory.IsFiniteMeasure μ] (hs : MeasureTheory.NullMeasurableSet s μ) (hs₀ : μ s ≠ 0) (hsc₀ : μ sᶜ ≠ 0)
Goal: ⨍⁻ (x : α), f x ∂μ ∈ openSegment ℝ≥0∞ (⨍⁻ (x : α) in s, f x ∂μ) (⨍⁻ (x : α) in sᶜ, f x ∂μ)
-- MeasureTheory.laverage_const
@[simp]
theorem laverage_const (μ : Measure α) [IsFiniteMeasure μ] [h : NeZero μ] (c : ℝ≥0∞) :
    ⨍⁻ _x, c ∂μ = c 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [h : NeZero μ] (c : ℝ≥0∞)
Goal: ⨍⁻ (_x : α), c ∂μ = c
-- MeasureTheory.setLaverage_const
theorem setLaverage_const (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) (c : ℝ≥0∞) : ⨍⁻ _x in s, c ∂μ = c 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ⊤) (c : ℝ≥0∞)
Goal: ⨍⁻ (_x : α) in s, c ∂μ = c
-- MeasureTheory.laverage_one
theorem laverage_one [IsFiniteMeasure μ] [NeZero μ] : ⨍⁻ _x, (1 : ℝ≥0∞) ∂μ = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] [NeZero μ]
Goal: ⨍⁻ (_x : α), 1 ∂μ = 1
-- MeasureTheory.setLaverage_one
theorem setLaverage_one (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : ⨍⁻ _x in s, (1 : ℝ≥0∞) ∂μ = 1 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (hs₀ : μ s ≠ 0) (hs : μ s ≠ ⊤)
Goal: ⨍⁻ (_x : α) in s, 1 ∂μ = 1
-- MeasureTheory.lintegral_laverage
theorem lintegral_laverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
    ∫⁻ _x, ⨍⁻ a, f a ∂μ ∂μ = ∫⁻ x, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ℝ≥0∞)
Goal: ∫⁻ (_x : α), ⨍⁻ (a : α), f a ∂μ ∂μ = ∫⁻ (x : α), f x ∂μ
-- MeasureTheory.setLintegral_setLaverage
theorem setLintegral_setLaverage (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α) :
    ∫⁻ _x in s, ⨍⁻ a in s, f a ∂μ ∂μ = ∫⁻ x in s, f x ∂μ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (f : α → ℝ≥0∞) (s : Set α)
Goal: ∫⁻ (_x : α) in s, ⨍⁻ (a : α) in s, f a ∂μ ∂μ = ∫⁻ (x : α) in s, f x ∂μ
-- MeasureTheory.average_zero
@[simp]
theorem average_zero : ⨍ _, (0 : E) ∂μ = 0 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] [NormedSpace ℝ E] (μ : MeasureTheory.Measure α)
Goal: ⨍ (x : α), 0 ∂μ = 0