ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IntervalIntegrable.mono_set'
theorem mono_set' (hf : IntervalIntegrable f μ a b) (hsub : Ι c d ⊆ Ι a b) :
    IntervalIntegrable f μ c d 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b c d : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (hsub : Ι c d ⊆ Ι a b)
Goal: IntervalIntegrable f μ c d
-- IntervalIntegrable.mono_fun
theorem mono_fun [NormedAddCommGroup F] {g : ℝ → F} (hf : IntervalIntegrable f μ a b)
    (hgm : AEStronglyMeasurable g (μ.restrict (Ι a b)))
    (hle : (fun x => ‖g x‖) ≤ᵐ[μ.restrict (Ι a b)] fun x => ‖f x‖) : IntervalIntegrable g μ a b 
Arguments: {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [NormedAddCommGroup F] {g : ℝ → F} (hf : IntervalIntegrable f μ a b) (hgm : MeasureTheory.AEStronglyMeasurable g (μ.restrict (Ι a b))) (hle : (fun x => ‖g x‖) ≤ᵐ[μ.restrict (Ι a b)] fun x => ‖f x‖)
Goal: IntervalIntegrable g μ a b
-- IntervalIntegrable.mono_fun'
theorem mono_fun' {g : ℝ → ℝ} (hg : IntervalIntegrable g μ a b)
    (hfm : AEStronglyMeasurable f (μ.restrict (Ι a b)))
    (hle : (fun x => ‖f x‖) ≤ᵐ[μ.restrict (Ι a b)] g) : IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {g : ℝ → ℝ} (hg : IntervalIntegrable g μ a b) (hfm : MeasureTheory.AEStronglyMeasurable f (μ.restrict (Ι a b))) (hle : (fun x => ‖f x‖) ≤ᵐ[μ.restrict (Ι a b)] g)
Goal: IntervalIntegrable f μ a b
-- IntervalIntegrable.aestronglyMeasurable
protected theorem aestronglyMeasurable (h : IntervalIntegrable f μ a b) :
    AEStronglyMeasurable f (μ.restrict (Ioc a b)) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc a b))
-- IntervalIntegrable.aestronglyMeasurable'
protected theorem aestronglyMeasurable' (h : IntervalIntegrable f μ a b) :
    AEStronglyMeasurable f (μ.restrict (Ioc b a)) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc b a))
-- IntervalIntegrable.smul
theorem smul [NormedField 𝕜] [NormedSpace 𝕜 E] {f : ℝ → E} {a b : ℝ} {μ : Measure ℝ}
    (h : IntervalIntegrable f μ a b) (r : 𝕜) : IntervalIntegrable (r • f) μ a b 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [NormedAddCommGroup E] [NormedField 𝕜] [NormedSpace 𝕜 E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b) (r : 𝕜)
Goal: IntervalIntegrable (r • f) μ a b
-- IntervalIntegrable.add
@[simp]
theorem add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) :
    IntervalIntegrable (fun x => f x + g x) μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f g : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)
Goal: IntervalIntegrable (fun x => f x + g x) μ a b
-- IntervalIntegrable.sub
@[simp]
theorem sub (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) :
    IntervalIntegrable (fun x => f x - g x) μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f g : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)
Goal: IntervalIntegrable (fun x => f x - g x) μ a b
-- IntervalIntegrable.sum
theorem sum (s : Finset ι) {f : ι → ℝ → E} (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) :
    IntervalIntegrable (∑ i ∈ s, f i) μ a b 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (s : Finset ι) {f : ι → ℝ → E} (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b)
Goal: IntervalIntegrable (∑ i ∈ s, f i) μ a b
-- IntervalIntegrable.mul_continuousOn
theorem mul_continuousOn {f g : ℝ → A} (hf : IntervalIntegrable f μ a b)
    (hg : ContinuousOn g [[a, b]]) : IntervalIntegrable (fun x => f x * g x) μ a b 
Arguments: {A : Type u_5} [NormedRing A] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f g : ℝ → A} (hf : IntervalIntegrable f μ a b) (hg : ContinuousOn g [[a, b]])
Goal: IntervalIntegrable (fun x => f x * g x) μ a b
-- IntervalIntegrable.continuousOn_mul
theorem continuousOn_mul {f g : ℝ → A} (hf : IntervalIntegrable f μ a b)
    (hg : ContinuousOn g [[a, b]]) : IntervalIntegrable (fun x => g x * f x) μ a b 
Arguments: {A : Type u_5} [NormedRing A] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f g : ℝ → A} (hf : IntervalIntegrable f μ a b) (hg : ContinuousOn g [[a, b]])
Goal: IntervalIntegrable (fun x => g x * f x) μ a b
-- IntervalIntegrable.const_mul
@[simp]
theorem const_mul {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A) :
    IntervalIntegrable (fun x => c * f x) μ a b 
Arguments: {A : Type u_5} [NormedRing A] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A)
Goal: IntervalIntegrable (fun x => c * f x) μ a b
-- IntervalIntegrable.mul_const
@[simp]
theorem mul_const {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A) :
    IntervalIntegrable (fun x => f x * c) μ a b 
Arguments: {A : Type u_5} [NormedRing A] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → A} (hf : IntervalIntegrable f μ a b) (c : A)
Goal: IntervalIntegrable (fun x => f x * c) μ a b
-- IntervalIntegrable.div_const
@[simp]
theorem div_const {𝕜 : Type*} {f : ℝ → 𝕜} [NormedField 𝕜] (h : IntervalIntegrable f μ a b)
    (c : 𝕜) : IntervalIntegrable (fun x => f x / c) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {𝕜 : Type u_6} {f : ℝ → 𝕜} [NormedField 𝕜] (h : IntervalIntegrable f μ a b) (c : 𝕜)
Goal: IntervalIntegrable (fun x => f x / c) μ a b
-- IntervalIntegrable.comp_mul_left
theorem comp_mul_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (c * x)) volume (a / c) (b / c) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (c * x)) MeasureTheory.volume (a / c) (b / c)
-- IntervalIntegrable.comp_mul_left_iff
theorem comp_mul_left_iff {c : ℝ} (hc : c ≠ 0) :
    IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c) ↔
      IntervalIntegrable f volume a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b c : ℝ} (hc : c ≠ 0)
Goal: IntervalIntegrable (fun x => f (c * x)) MeasureTheory.volume (a / c) (b / c) ↔
  IntervalIntegrable f MeasureTheory.volume a b
-- IntervalIntegrable.comp_mul_right
theorem comp_mul_right (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (x * c)) volume (a / c) (b / c) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (x * c)) MeasureTheory.volume (a / c) (b / c)
-- IntervalIntegrable.comp_add_right
theorem comp_add_right (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (x + c)) MeasureTheory.volume (a - c) (b - c)
-- IntervalIntegrable.comp_add_left
theorem comp_add_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (c + x)) volume (a - c) (b - c) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (c + x)) MeasureTheory.volume (a - c) (b - c)
-- IntervalIntegrable.comp_sub_right
theorem comp_sub_right (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (x - c)) volume (a + c) (b + c) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (x - c)) MeasureTheory.volume (a + c) (b + c)
-- IntervalIntegrable.iff_comp_neg
theorem iff_comp_neg :
    IntervalIntegrable f volume a b ↔ IntervalIntegrable (fun x => f (-x)) volume (-a) (-b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ}
Goal: IntervalIntegrable f MeasureTheory.volume a b ↔ IntervalIntegrable (fun x => f (-x)) MeasureTheory.volume (-a) (-b)
-- IntervalIntegrable.comp_sub_left
theorem comp_sub_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
    IntervalIntegrable (fun x => f (c - x)) volume (c - a) (c - b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (c : ℝ)
Goal: IntervalIntegrable (fun x => f (c - x)) MeasureTheory.volume (c - a) (c - b)
-- ContinuousOn.intervalIntegrable
theorem ContinuousOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : ContinuousOn u (uIcc a b)) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {u : ℝ → E} {a b : ℝ} (hu : ContinuousOn u [[a, b]])
Goal: IntervalIntegrable u μ a b
-- ContinuousOn.intervalIntegrable_of_Icc
theorem ContinuousOn.intervalIntegrable_of_Icc {u : ℝ → E} {a b : ℝ} (h : a ≤ b)
    (hu : ContinuousOn u (Icc a b)) : IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {u : ℝ → E} {a b : ℝ} (h : a ≤ b) (hu : ContinuousOn u (Set.Icc a b))
Goal: IntervalIntegrable u μ a b
-- Continuous.intervalIntegrable
theorem Continuous.intervalIntegrable {u : ℝ → E} (hu : Continuous u) (a b : ℝ) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {u : ℝ → E} (hu : Continuous u) (a b : ℝ)
Goal: IntervalIntegrable u μ a b
-- MonotoneOn.intervalIntegrable
theorem MonotoneOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : MonotoneOn u (uIcc a b)) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] [ConditionallyCompleteLinearOrder E] [OrderTopology E] [SecondCountableTopology E] {u : ℝ → E} {a b : ℝ} (hu : MonotoneOn u [[a, b]])
Goal: IntervalIntegrable u μ a b
-- AntitoneOn.intervalIntegrable
theorem AntitoneOn.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : AntitoneOn u (uIcc a b)) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] [ConditionallyCompleteLinearOrder E] [OrderTopology E] [SecondCountableTopology E] {u : ℝ → E} {a b : ℝ} (hu : AntitoneOn u [[a, b]])
Goal: IntervalIntegrable u μ a b
-- Monotone.intervalIntegrable
theorem Monotone.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : Monotone u) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] [ConditionallyCompleteLinearOrder E] [OrderTopology E] [SecondCountableTopology E] {u : ℝ → E} {a b : ℝ} (hu : Monotone u)
Goal: IntervalIntegrable u μ a b
-- Antitone.intervalIntegrable
theorem Antitone.intervalIntegrable {u : ℝ → E} {a b : ℝ} (hu : Antitone u) :
    IntervalIntegrable u μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] [ConditionallyCompleteLinearOrder E] [OrderTopology E] [SecondCountableTopology E] {u : ℝ → E} {a b : ℝ} (hu : Antitone u)
Goal: IntervalIntegrable u μ a b
-- Filter.Tendsto.eventually_intervalIntegrable_ae
theorem Filter.Tendsto.eventually_intervalIntegrable_ae {f : ℝ → E} {μ : Measure ℝ}
    {l l' : Filter ℝ} (hfm : StronglyMeasurableAtFilter f l' μ) [TendstoIxxClass Ioc l l']
    [IsMeasurablyGenerated l'] (hμ : μ.FiniteAtFilter l') {c : E} (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c))
    {u v : ι → ℝ} {lt : Filter ι} (hu : Tendsto u lt l) (hv : Tendsto v lt l) :
    ∀ᶠ t in lt, IntervalIntegrable f μ (u t) (v t) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {l l' : Filter ℝ} (hfm : StronglyMeasurableAtFilter f l' μ) [Filter.TendstoIxxClass Set.Ioc l l'] [l'.IsMeasurablyGenerated] (hμ : μ.FiniteAtFilter l') {c : E} (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) {u v : ι → ℝ} {lt : Filter ι} (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l)
Goal: ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (u t) (v t)
-- Filter.Tendsto.eventually_intervalIntegrable
theorem Filter.Tendsto.eventually_intervalIntegrable {f : ℝ → E} {μ : Measure ℝ} {l l' : Filter ℝ}
    (hfm : StronglyMeasurableAtFilter f l' μ) [TendstoIxxClass Ioc l l'] [IsMeasurablyGenerated l']
    (hμ : μ.FiniteAtFilter l') {c : E} (hf : Tendsto f l' (𝓝 c)) {u v : ι → ℝ} {lt : Filter ι}
    (hu : Tendsto u lt l) (hv : Tendsto v lt l) : ∀ᶠ t in lt, IntervalIntegrable f μ (u t) (v t) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {l l' : Filter ℝ} (hfm : StronglyMeasurableAtFilter f l' μ) [Filter.TendstoIxxClass Set.Ioc l l'] [l'.IsMeasurablyGenerated] (hμ : μ.FiniteAtFilter l') {c : E} (hf : Filter.Tendsto f l' (𝓝 c)) {u v : ι → ℝ} {lt : Filter ι} (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l)
Goal: ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (u t) (v t)
-- intervalIntegral.integral_zero
@[simp]
theorem integral_zero : (∫ _ in a..b, (0 : E) ∂μ) = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ}
Goal: ∫ (x : ℝ) in a..b, 0 ∂μ = 0
-- intervalIntegral.integral_of_le
theorem integral_of_le (h : a ≤ b) : ∫ x in a..b, f x ∂μ = ∫ x in Ioc a b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : a ≤ b)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in Set.Ioc a b, f x ∂μ
-- intervalIntegral.integral_same
@[simp]
theorem integral_same : ∫ x in a..a, f x ∂μ = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ}
Goal: ∫ (x : ℝ) in a..a, f x ∂μ = 0
-- intervalIntegral.integral_symm
theorem integral_symm (a b) : ∫ x in b..a, f x ∂μ = -∫ x in a..b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (a b : ℝ)
Goal: ∫ (x : ℝ) in b..a, f x ∂μ = -∫ (x : ℝ) in a..b, f x ∂μ
-- intervalIntegral.integral_of_ge
theorem integral_of_ge (h : b ≤ a) : ∫ x in a..b, f x ∂μ = -∫ x in Ioc b a, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : b ≤ a)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = -∫ (x : ℝ) in Set.Ioc b a, f x ∂μ
-- intervalIntegral.intervalIntegral_eq_integral_uIoc
theorem intervalIntegral_eq_integral_uIoc (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) :
    ∫ x in a..b, f x ∂μ = (if a ≤ b then 1 else -1 : ℝ) • ∫ x in Ι a b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (a b : ℝ) (μ : MeasureTheory.Measure ℝ)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = (if a ≤ b then 1 else -1) • ∫ (x : ℝ) in Ι a b, f x ∂μ
-- intervalIntegral.norm_intervalIntegral_eq
theorem norm_intervalIntegral_eq (f : ℝ → E) (a b : ℝ) (μ : Measure ℝ) :
    ‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E) (a b : ℝ) (μ : MeasureTheory.Measure ℝ)
Goal: ‖∫ (x : ℝ) in a..b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖
-- intervalIntegral.abs_intervalIntegral_eq
theorem abs_intervalIntegral_eq (f : ℝ → ℝ) (a b : ℝ) (μ : Measure ℝ) :
    |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| 
Arguments: (f : ℝ → ℝ) (a b : ℝ) (μ : MeasureTheory.Measure ℝ)
Goal: |∫ (x : ℝ) in a..b, f x ∂μ| = |∫ (x : ℝ) in Ι a b, f x ∂μ|
-- intervalIntegral.integral_cases
theorem integral_cases (f : ℝ → E) (a b) :
    (∫ x in a..b, f x ∂μ) ∈ ({∫ x in Ι a b, f x ∂μ, -∫ x in Ι a b, f x ∂μ} : Set E) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} (f : ℝ → E) (a b : ℝ)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ ∈ {∫ (x : ℝ) in Ι a b, f x ∂μ, -∫ (x : ℝ) in Ι a b, f x ∂μ}
-- intervalIntegral.integral_undef
nonrec theorem integral_undef (h : ¬IntervalIntegrable f μ a b) : ∫ x in a..b, f x ∂μ = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : ¬IntervalIntegrable f μ a b)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0
-- intervalIntegral.intervalIntegrable_of_integral_ne_zero
theorem intervalIntegrable_of_integral_ne_zero {a b : ℝ} {f : ℝ → E} {μ : Measure ℝ}
    (h : (∫ x in a..b, f x ∂μ) ≠ 0) : IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : ∫ (x : ℝ) in a..b, f x ∂μ ≠ 0)
Goal: IntervalIntegrable f μ a b
-- intervalIntegral.integral_non_aestronglyMeasurable
nonrec theorem integral_non_aestronglyMeasurable
    (hf : ¬AEStronglyMeasurable f (μ.restrict (Ι a b))) :
    ∫ x in a..b, f x ∂μ = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hf : ¬MeasureTheory.AEStronglyMeasurable f (μ.restrict (Ι a b)))
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0
-- intervalIntegral.integral_non_aestronglyMeasurable_of_le
theorem integral_non_aestronglyMeasurable_of_le (h : a ≤ b)
    (hf : ¬AEStronglyMeasurable f (μ.restrict (Ioc a b))) : ∫ x in a..b, f x ∂μ = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : a ≤ b) (hf : ¬MeasureTheory.AEStronglyMeasurable f (μ.restrict (Set.Ioc a b)))
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0
-- intervalIntegral.norm_integral_min_max
theorem norm_integral_min_max (f : ℝ → E) :
    ‖∫ x in min a b..max a b, f x ∂μ‖ = ‖∫ x in a..b, f x ∂μ‖ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (f : ℝ → E)
Goal: ‖∫ (x : ℝ) in min a b..max a b, f x ∂μ‖ = ‖∫ (x : ℝ) in a..b, f x ∂μ‖
-- intervalIntegral.norm_integral_eq_norm_integral_Ioc
theorem norm_integral_eq_norm_integral_Ioc (f : ℝ → E) :
    ‖∫ x in a..b, f x ∂μ‖ = ‖∫ x in Ι a b, f x ∂μ‖ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (f : ℝ → E)
Goal: ‖∫ (x : ℝ) in a..b, f x ∂μ‖ = ‖∫ (x : ℝ) in Ι a b, f x ∂μ‖
-- intervalIntegral.abs_integral_eq_abs_integral_uIoc
theorem abs_integral_eq_abs_integral_uIoc (f : ℝ → ℝ) :
    |∫ x in a..b, f x ∂μ| = |∫ x in Ι a b, f x ∂μ| 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (f : ℝ → ℝ)
Goal: |∫ (x : ℝ) in a..b, f x ∂μ| = |∫ (x : ℝ) in Ι a b, f x ∂μ|
-- intervalIntegral.norm_integral_le_integral_norm_Ioc
theorem norm_integral_le_integral_norm_Ioc : ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in Ι a b, ‖f x‖ ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ}
Goal: ‖∫ (x : ℝ) in a..b, f x ∂μ‖ ≤ ∫ (x : ℝ) in Ι a b, ‖f x‖ ∂μ
-- intervalIntegral.norm_integral_le_abs_integral_norm
theorem norm_integral_le_abs_integral_norm : ‖∫ x in a..b, f x ∂μ‖ ≤ |∫ x in a..b, ‖f x‖ ∂μ| 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ}
Goal: ‖∫ (x : ℝ) in a..b, f x ∂μ‖ ≤ |∫ (x : ℝ) in a..b, ‖f x‖ ∂μ|
-- intervalIntegral.norm_integral_le_integral_norm
theorem norm_integral_le_integral_norm (h : a ≤ b) :
    ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in a..b, ‖f x‖ ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h : a ≤ b)
Goal: ‖∫ (x : ℝ) in a..b, f x ∂μ‖ ≤ ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ
-- intervalIntegral.norm_integral_le_of_norm_le
nonrec theorem norm_integral_le_of_norm_le {g : ℝ → ℝ} (h : ∀ᵐ t ∂μ.restrict <| Ι a b, ‖f t‖ ≤ g t)
    (hbound : IntervalIntegrable g μ a b) : ‖∫ t in a..b, f t ∂μ‖ ≤ |∫ t in a..b, g t ∂μ| 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {g : ℝ → ℝ} (h : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖f t‖ ≤ g t) (hbound : IntervalIntegrable g μ a b)
Goal: ‖∫ (t : ℝ) in a..b, f t ∂μ‖ ≤ |∫ (t : ℝ) in a..b, g t ∂μ|
-- intervalIntegral.norm_integral_le_of_norm_le_const_ae
theorem norm_integral_le_of_norm_le_const_ae {a b C : ℝ} {f : ℝ → E}
    (h : ∀ᵐ x, x ∈ Ι a b → ‖f x‖ ≤ C) : ‖∫ x in a..b, f x‖ ≤ C * |b - a| 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b C : ℝ} {f : ℝ → E} (h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C)
Goal: ‖∫ (x : ℝ) in a..b, f x‖ ≤ C * |b - a|
-- intervalIntegral.norm_integral_le_of_norm_le_const
theorem norm_integral_le_of_norm_le_const {a b C : ℝ} {f : ℝ → E} (h : ∀ x ∈ Ι a b, ‖f x‖ ≤ C) :
    ‖∫ x in a..b, f x‖ ≤ C * |b - a| 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b C : ℝ} {f : ℝ → E} (h : ∀ x ∈ Ι a b, ‖f x‖ ≤ C)
Goal: ‖∫ (x : ℝ) in a..b, f x‖ ≤ C * |b - a|
-- intervalIntegral.integral_add
@[simp]
nonrec theorem integral_add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) :
    ∫ x in a..b, f x + g x ∂μ = (∫ x in a..b, f x ∂μ) + ∫ x in a..b, g x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f g : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)
Goal: ∫ (x : ℝ) in a..b, f x + g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_finset_sum
nonrec theorem integral_finset_sum {ι} {s : Finset ι} {f : ι → ℝ → E}
    (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) :
    ∫ x in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ x in a..b, f i x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {ι : Type u_6} {s : Finset ι} {f : ι → ℝ → E} (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b)
Goal: ∫ (x : ℝ) in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ (x : ℝ) in a..b, f i x ∂μ
-- intervalIntegral.integral_neg
@[simp]
nonrec theorem integral_neg : ∫ x in a..b, -f x ∂μ = -∫ x in a..b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ}
Goal: ∫ (x : ℝ) in a..b, -f x ∂μ = -∫ (x : ℝ) in a..b, f x ∂μ
-- intervalIntegral.integral_sub
@[simp]
theorem integral_sub (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) :
    ∫ x in a..b, f x - g x ∂μ = (∫ x in a..b, f x ∂μ) - ∫ x in a..b, g x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {f g : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)
Goal: ∫ (x : ℝ) in a..b, f x - g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_smul
@[simp]
nonrec theorem integral_smul {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    [SMulCommClass ℝ 𝕜 E] (r : 𝕜) (f : ℝ → E) :
    ∫ x in a..b, r • f x ∂μ = r • ∫ x in a..b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {𝕜 : Type u_6} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℝ 𝕜 E] (r : 𝕜) (f : ℝ → E)
Goal: ∫ (x : ℝ) in a..b, r • f x ∂μ = r • ∫ (x : ℝ) in a..b, f x ∂μ
-- intervalIntegral.integral_smul_const
@[simp]
nonrec theorem integral_smul_const {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : ℝ → 𝕜) (c : E) :
    ∫ x in a..b, f x • c ∂μ = (∫ x in a..b, f x ∂μ) • c 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {𝕜 : Type u_6} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : ℝ → 𝕜) (c : E)
Goal: ∫ (x : ℝ) in a..b, f x • c ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) • c
-- intervalIntegral.integral_const_mul
@[simp]
theorem integral_const_mul {𝕜 : Type*} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :
    ∫ x in a..b, r * f x ∂μ = r * ∫ x in a..b, f x ∂μ 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {𝕜 : Type u_6} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜)
Goal: ∫ (x : ℝ) in a..b, r * f x ∂μ = r * ∫ (x : ℝ) in a..b, f x ∂μ