ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.integral_deriv_mul_eq_sub_of_hasDeriv_right
theorem integral_deriv_mul_eq_sub_of_hasDeriv_right {u v u' v' : ℝ → A}
    (hu : ContinuousOn u [[a, b]])
    (hv : ContinuousOn v [[a, b]])
    (huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x)
    (hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ContinuousOn u [[a, b]]) (hv : ContinuousOn v [[a, b]]) (huu' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Set.Ioi x) x) (hvv' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Set.Ioi x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a
-- intervalIntegral.integral_deriv_mul_eq_sub_of_hasDerivAt
theorem integral_deriv_mul_eq_sub_of_hasDerivAt {u v u' v' : ℝ → A}
    (hu : ContinuousOn u [[a, b]])
    (hv : ContinuousOn v [[a, b]])
    (huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt u (u' x) x)
    (hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt v (v' x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ContinuousOn u [[a, b]]) (hv : ContinuousOn v [[a, b]]) (huu' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt u (u' x) x) (hvv' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt v (v' x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a
-- intervalIntegral.integral_deriv_mul_eq_sub_of_hasDerivWithinAt
theorem integral_deriv_mul_eq_sub_of_hasDerivWithinAt {u v u' v' : ℝ → A}
    (hu : ∀ x ∈ [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)
    (hv : ∀ x ∈ [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ∀ x ∈ [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x) (hv : ∀ x ∈ [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a
-- intervalIntegral.integral_deriv_mul_eq_sub
theorem integral_deriv_mul_eq_sub {u v u' v' : ℝ → A}
    (hu : ∀ x ∈ [[a, b]], HasDerivAt u (u' x) x)
    (hv : ∀ x ∈ [[a, b]], HasDerivAt v (v' x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ∀ x ∈ [[a, b]], HasDerivAt u (u' x) x) (hv : ∀ x ∈ [[a, b]], HasDerivAt v (v' x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a
-- intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right
theorem integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right {u v u' v' : ℝ → A}
    (hu : ContinuousOn u [[a, b]])
    (hv : ContinuousOn v [[a, b]])
    (huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x)
    (hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, u' x * v x 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ContinuousOn u [[a, b]]) (hv : ContinuousOn v [[a, b]]) (huu' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Set.Ioi x) x) (hvv' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Set.Ioi x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x
-- intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivAt
theorem integral_mul_deriv_eq_deriv_mul_of_hasDerivAt {u v u' v' : ℝ → A}
    (hu : ContinuousOn u [[a, b]])
    (hv : ContinuousOn v [[a, b]])
    (huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt u (u' x) x)
    (hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt v (v' x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, u' x * v x 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ContinuousOn u [[a, b]]) (hv : ContinuousOn v [[a, b]]) (huu' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt u (u' x) x) (hvv' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt v (v' x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x
-- intervalIntegral.integral_mul_deriv_eq_deriv_mul_of_hasDerivWithinAt
theorem integral_mul_deriv_eq_deriv_mul_of_hasDerivWithinAt {u v u' v' : ℝ → A}
    (hu : ∀ x ∈ [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x)
    (hv : ∀ x ∈ [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, u' x * v x 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ∀ x ∈ [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x) (hv : ∀ x ∈ [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x
-- intervalIntegral.integral_mul_deriv_eq_deriv_mul
theorem integral_mul_deriv_eq_deriv_mul {u v u' v' : ℝ → A}
    (hu : ∀ x ∈ [[a, b]], HasDerivAt u (u' x) x)
    (hv : ∀ x ∈ [[a, b]], HasDerivAt v (v' x) x)
    (hu' : IntervalIntegrable u' volume a b)
    (hv' : IntervalIntegrable v' volume a b) :
    ∫ x in a..b, u x * v' x = u b * v b - u a * v a - ∫ x in a..b, u' x * v x 
Arguments: {A : Type u_5} {a b : ℝ} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {u v u' v' : ℝ → A} (hu : ∀ x ∈ [[a, b]], HasDerivAt u (u' x) x) (hv : ∀ x ∈ [[a, b]], HasDerivAt v (v' x) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x
-- intervalIntegral.integral_comp_smul_deriv'''
theorem integral_comp_smul_deriv''' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
    (hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]]) :
    (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u 
Arguments: {a b : ℝ} {G : Type u_6} [NormedAddCommGroup G] [NormedSpace ℝ G] {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hg_cont : ContinuousOn g (f '' Set.Ioo (min a b) (max a b))) (hg1 : MeasureTheory.IntegrableOn g (f '' [[a, b]]) MeasureTheory.volume) (hg2 : MeasureTheory.IntegrableOn (fun x => f' x • (g ∘ f) x) [[a, b]] MeasureTheory.volume)
Goal: ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
-- intervalIntegral.integral_comp_smul_deriv''
theorem integral_comp_smul_deriv'' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]])) :
    (∫ x in a..b, f' x • (g ∘ f) x) = ∫ u in f a..f b, g u 
Arguments: {a b : ℝ} {G : Type u_6} [NormedAddCommGroup G] [NormedSpace ℝ G] {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u
-- intervalIntegral.integral_comp_smul_deriv'
theorem integral_comp_smul_deriv' {f f' : ℝ → ℝ} {g : ℝ → G}
    (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (h' : ContinuousOn f' (uIcc a b))
    (hg : ContinuousOn g (f '' [[a, b]])) :
    (∫ x in a..b, f' x • (g ∘ f) x) = ∫ x in f a..f b, g x 
Arguments: {a b : ℝ} {G : Type u_6} [NormedAddCommGroup G] [NormedSpace ℝ G] {f f' : ℝ → ℝ} {g : ℝ → G} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (h' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (x : ℝ) in f a..f b, g x
-- intervalIntegral.integral_comp_smul_deriv
theorem integral_comp_smul_deriv {f f' : ℝ → ℝ} {g : ℝ → G}
    (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (h' : ContinuousOn f' (uIcc a b))
    (hg : Continuous g) : (∫ x in a..b, f' x • (g ∘ f) x) = ∫ x in f a..f b, g x 
Arguments: {a b : ℝ} {G : Type u_6} [NormedAddCommGroup G] [NormedSpace ℝ G] {f f' : ℝ → ℝ} {g : ℝ → G} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (h' : ContinuousOn f' [[a, b]]) (hg : Continuous g)
Goal: ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (x : ℝ) in f a..f b, g x
-- intervalIntegral.integral_deriv_comp_smul_deriv'
theorem integral_deriv_comp_smul_deriv' {f f' : ℝ → ℝ} {g g' : ℝ → E} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g [[f a, f b]])
    (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x)
    (hg' : ContinuousOn g' (f '' [[a, b]])) :
    (∫ x in a..b, f' x • (g' ∘ f) x) = (g ∘ f) b - (g ∘ f) a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → ℝ} {g g' : ℝ → E} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g [[f a, f b]]) (hgg' : ∀ x ∈ Set.Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Set.Ioi x) x) (hg' : ContinuousOn g' (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a
-- intervalIntegral.integral_deriv_comp_smul_deriv
theorem integral_deriv_comp_smul_deriv {f f' : ℝ → ℝ} {g g' : ℝ → E}
    (hf : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (hg : ∀ x ∈ uIcc a b, HasDerivAt g (g' (f x)) (f x)) (hf' : ContinuousOn f' (uIcc a b))
    (hg' : Continuous g') : (∫ x in a..b, f' x • (g' ∘ f) x) = (g ∘ f) b - (g ∘ f) a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → ℝ} {g g' : ℝ → E} (hf : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (hg : ∀ x ∈ [[a, b]], HasDerivAt g (g' (f x)) (f x)) (hf' : ContinuousOn f' [[a, b]]) (hg' : Continuous g')
Goal: ∫ (x : ℝ) in a..b, f' x • (g' ∘ f) x = (g ∘ f) b - (g ∘ f) a
-- intervalIntegral.integral_comp_mul_deriv'''
theorem integral_comp_mul_deriv''' {a b : ℝ} {f f' : ℝ → ℝ} {g : ℝ → ℝ}
    (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
    (hg2 : IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]]) :
    (∫ x in a..b, (g ∘ f) x * f' x) = ∫ u in f a..f b, g u 
Arguments: {a b : ℝ} {f f' g : ℝ → ℝ} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hg_cont : ContinuousOn g (f '' Set.Ioo (min a b) (max a b))) (hg1 : MeasureTheory.IntegrableOn g (f '' [[a, b]]) MeasureTheory.volume) (hg2 : MeasureTheory.IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]] MeasureTheory.volume)
Goal: ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u
-- intervalIntegral.integral_comp_mul_deriv''
theorem integral_comp_mul_deriv'' {f f' g : ℝ → ℝ} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]])) :
    (∫ x in a..b, (g ∘ f) x * f' x) = ∫ u in f a..f b, g u 
Arguments: {a b : ℝ} {f f' g : ℝ → ℝ} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u
-- intervalIntegral.integral_comp_mul_deriv'
theorem integral_comp_mul_deriv' {f f' g : ℝ → ℝ} (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (h' : ContinuousOn f' (uIcc a b)) (hg : ContinuousOn g (f '' [[a, b]])) :
    (∫ x in a..b, (g ∘ f) x * f' x) = ∫ x in f a..f b, g x 
Arguments: {a b : ℝ} {f f' g : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (h' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x
-- intervalIntegral.integral_comp_mul_deriv
theorem integral_comp_mul_deriv {f f' g : ℝ → ℝ} (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (h' : ContinuousOn f' (uIcc a b)) (hg : Continuous g) :
    (∫ x in a..b, (g ∘ f) x * f' x) = ∫ x in f a..f b, g x 
Arguments: {a b : ℝ} {f f' g : ℝ → ℝ} (h : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (h' : ContinuousOn f' [[a, b]]) (hg : Continuous g)
Goal: ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x
-- intervalIntegral.integral_deriv_comp_mul_deriv'
theorem integral_deriv_comp_mul_deriv' {f f' g g' : ℝ → ℝ} (hf : ContinuousOn f [[a, b]])
    (hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g [[f a, f b]])
    (hgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x)
    (hg' : ContinuousOn g' (f '' [[a, b]])) :
    (∫ x in a..b, (g' ∘ f) x * f' x) = (g ∘ f) b - (g ∘ f) a 
Arguments: {a b : ℝ} {f f' g g' : ℝ → ℝ} (hf : ContinuousOn f [[a, b]]) (hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hf' : ContinuousOn f' [[a, b]]) (hg : ContinuousOn g [[f a, f b]]) (hgg' : ∀ x ∈ Set.Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Set.Ioi x) x) (hg' : ContinuousOn g' (f '' [[a, b]]))
Goal: ∫ (x : ℝ) in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a
-- intervalIntegral.integral_deriv_comp_mul_deriv
theorem integral_deriv_comp_mul_deriv {f f' g g' : ℝ → ℝ}
    (hf : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (hg : ∀ x ∈ uIcc a b, HasDerivAt g (g' (f x)) (f x)) (hf' : ContinuousOn f' (uIcc a b))
    (hg' : Continuous g') : (∫ x in a..b, (g' ∘ f) x * f' x) = (g ∘ f) b - (g ∘ f) a 
Arguments: {a b : ℝ} {f f' g g' : ℝ → ℝ} (hf : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (hg : ∀ x ∈ [[a, b]], HasDerivAt g (g' (f x)) (f x)) (hf' : ContinuousOn f' [[a, b]]) (hg' : Continuous g')
Goal: ∫ (x : ℝ) in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a
-- Besicovitch.SatelliteConfig.centerAndRescale_center
theorem centerAndRescale_center : a.centerAndRescale.c (last N) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ)
Goal: a.centerAndRescale.c (Fin.last N) = 0
-- Besicovitch.SatelliteConfig.centerAndRescale_radius
theorem centerAndRescale_radius {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) :
    a.centerAndRescale.r (last N) = 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ)
Goal: a.centerAndRescale.r (Fin.last N) = 1
-- Besicovitch.card_le_of_separated
theorem card_le_of_separated (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ 5 ^ finrank ℝ E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (s : Finset E) (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖)
Goal: s.card ≤ 5 ^ FiniteDimensional.finrank ℝ E
-- Besicovitch.multiplicity_le
theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: Besicovitch.multiplicity E ≤ 5 ^ FiniteDimensional.finrank ℝ E
-- Besicovitch.card_le_multiplicity
theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖)
Goal: s.card ≤ Besicovitch.multiplicity E
-- Besicovitch.exists_goodδ
theorem exists_goodδ :
    ∃ δ : ℝ, 0 < δ ∧ δ < 1 ∧ ∀ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ 2) →
      (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: ∃ δ,
  0 < δ ∧
    δ < 1 ∧
      ∀ (s : Finset E),
        (∀ c ∈ s, ‖c‖ ≤ 2) → (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ Besicovitch.multiplicity E
-- Besicovitch.goodδ_lt_one
theorem goodδ_lt_one : goodδ E < 1 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: Besicovitch.goodδ E < 1
-- Besicovitch.one_lt_goodτ
theorem one_lt_goodτ : 1 < goodτ E 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: 1 < Besicovitch.goodτ E
-- Besicovitch.card_le_multiplicity_of_δ
theorem card_le_multiplicity_of_δ {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
    (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - goodδ E ≤ ‖c - d‖) : s.card ≤ multiplicity E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2) (h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - Besicovitch.goodδ E ≤ ‖c - d‖)
Goal: s.card ≤ Besicovitch.multiplicity E
-- Besicovitch.le_multiplicity_of_δ_of_fin
theorem le_multiplicity_of_δ_of_fin {n : ℕ} (f : Fin n → E) (h : ∀ i, ‖f i‖ ≤ 2)
    (h' : Pairwise fun i j => 1 - goodδ E ≤ ‖f i - f j‖) : n ≤ multiplicity E 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {n : ℕ} (f : Fin n → E) (h : ∀ (i : Fin n), ‖f i‖ ≤ 2) (h' : Pairwise fun i j => 1 - Besicovitch.goodδ E ≤ ‖f i - f j‖)
Goal: n ≤ Besicovitch.multiplicity E
-- Besicovitch.SatelliteConfig.exists_normalized_aux1
theorem exists_normalized_aux1 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1)
    (i j : Fin N.succ) (inej : i ≠ j) : 1 - δ ≤ ‖a.c i - a.c j‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ) (lastr : a.r (Fin.last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) (i j : Fin N.succ) (inej : i ≠ j)
Goal: 1 - δ ≤ ‖a.c i - a.c j‖
-- Besicovitch.SatelliteConfig.exists_normalized_aux2
theorem exists_normalized_aux2 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (lastc : a.c (last N) = 0) (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4)
    (hδ2 : δ ≤ 1) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖) :
    1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ) (lastc : a.c (Fin.last N) = 0) (lastr : a.r (Fin.last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) (i j : Fin N.succ) (inej : i ≠ j) (hi : ‖a.c i‖ ≤ 2) (hj : 2 < ‖a.c j‖)
Goal: 1 - δ ≤ ‖a.c i - (2 / ‖a.c j‖) • a.c j‖
-- Besicovitch.SatelliteConfig.exists_normalized_aux3
theorem exists_normalized_aux3 {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ)
    (lastc : a.c (last N) = 0) (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4)
    (i j : Fin N.succ) (inej : i ≠ j) (hi : 2 < ‖a.c i‖) (hij : ‖a.c i‖ ≤ ‖a.c j‖) :
    1 - δ ≤ ‖(2 / ‖a.c i‖) • a.c i - (2 / ‖a.c j‖) • a.c j‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ) (lastc : a.c (Fin.last N) = 0) (lastr : a.r (Fin.last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (i j : Fin N.succ) (inej : i ≠ j) (hi : 2 < ‖a.c i‖) (hij : ‖a.c i‖ ≤ ‖a.c j‖)
Goal: 1 - δ ≤ ‖(2 / ‖a.c i‖) • a.c i - (2 / ‖a.c j‖) • a.c j‖
-- Besicovitch.SatelliteConfig.exists_normalized
theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (lastc : a.c (last N) = 0)
    (lastr : a.r (last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1) :
    ∃ c' : Fin N.succ → E, (∀ n, ‖c' n‖ ≤ 2) ∧ Pairwise fun i j => 1 - δ ≤ ‖c' i - c' j‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {N : ℕ} {τ : ℝ} (a : Besicovitch.SatelliteConfig E N τ) (lastc : a.c (Fin.last N) = 0) (lastr : a.r (Fin.last N) = 1) (hτ : 1 ≤ τ) (δ : ℝ) (hδ1 : τ ≤ 1 + δ / 4) (hδ2 : δ ≤ 1)
Goal: ∃ c', (∀ (n : Fin N.succ), ‖c' n‖ ≤ 2) ∧ Pairwise fun i j => 1 - δ ≤ ‖c' i - c' j‖
-- Besicovitch.isEmpty_satelliteConfig_multiplicity
theorem isEmpty_satelliteConfig_multiplicity :
    IsEmpty (SatelliteConfig E (multiplicity E) (goodτ E)) 
Arguments: (E : Type u_1) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
Goal: IsEmpty (Besicovitch.SatelliteConfig E (Besicovitch.multiplicity E) (Besicovitch.goodτ E))
-- MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero
theorem tendsto_measure_symmDiff_preimage_nhds_zero
    {l : Filter α} {f : α → C(X, Y)} {g : C(X, Y)} {s : Set Y} (hfg : Tendsto f l (𝓝 g))
    (hf : ∀ᶠ a in l, MeasurePreserving (f a) μ ν) (hg : MeasurePreserving g μ ν)
    (hs : MeasurableSet s) (hνs : ν s ≠ ∞) :
    Tendsto (fun a ↦ μ ((f a ⁻¹' s) ∆ (g ⁻¹' s))) l (𝓝 0) 
Arguments: {α : Type u_1} {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] {l : Filter α} {f : α → C(X, Y)} {g : C(X, Y)} {s : Set Y} (hfg : Filter.Tendsto f l (𝓝 g)) (hf : ∀ᶠ (a : α) in l, MeasureTheory.MeasurePreserving (⇑(f a)) μ ν) (hg : MeasureTheory.MeasurePreserving (⇑g) μ ν) (hs : MeasurableSet s) (hνs : ν s ≠ ⊤)
Goal: Filter.Tendsto (fun a => μ ((⇑(f a) ⁻¹' s) ∆ (⇑g ⁻¹' s))) l (𝓝 0)
-- MeasureTheory.Lp.compMeasurePreserving_continuous
theorem compMeasurePreserving_continuous (hp : p ≠ ∞) :
    Continuous fun gf : Lp E p ν × {f : C(X, Y) // MeasurePreserving f μ ν} ↦
      compMeasurePreserving gf.2.1 gf.2.2 gf.1 
Arguments: {X : Type u_2} {Y : Type u_3} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X] [R1Space X] [TopologicalSpace Y] [MeasurableSpace Y] [BorelSpace Y] [R1Space Y] (μ : MeasureTheory.Measure X) (ν : MeasureTheory.Measure Y) [μ.InnerRegularCompactLTTop] [MeasureTheory.IsLocallyFiniteMeasure ν] (E : Type u_4) [NormedAddCommGroup E] {p : ℝ≥0∞} [Fact (1 ≤ p)] (hp : p ≠ ⊤)
Goal: Continuous fun gf => (MeasureTheory.Lp.compMeasurePreserving ⇑↑gf.2 ⋯) gf.1