ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.integral_mul_const
@[simp]
theorem integral_mul_const {𝕜 : Type*} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :
    ∫ x in a..b, f x * r ∂μ = (∫ x in a..b, f x ∂μ) * r 
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, f x * r ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) * r
-- intervalIntegral.integral_div
@[simp]
theorem integral_div {𝕜 : Type*} [RCLike 𝕜] (r : 𝕜) (f : ℝ → 𝕜) :
    ∫ x in a..b, f x / r ∂μ = (∫ x in a..b, f x ∂μ) / r 
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, f x / r ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) / r
-- intervalIntegral.integral_const'
theorem integral_const' (c : E) :
    ∫ _ in a..b, c ∂μ = ((μ <| Ioc a b).toReal - (μ <| Ioc b a).toReal) • c 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (c : E)
Goal: ∫ (x : ℝ) in a..b, c ∂μ = ((μ (Set.Ioc a b)).toReal - (μ (Set.Ioc b a)).toReal) • c
-- intervalIntegral.integral_const
@[simp]
theorem integral_const (c : E) : ∫ _ in a..b, c = (b - a) • c 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (c : E)
Goal: ∫ (x : ℝ) in a..b, c = (b - a) • c
-- intervalIntegral.integral_smul_measure
nonrec theorem integral_smul_measure (c : ℝ≥0∞) :
    ∫ x in a..b, f x ∂c • μ = c.toReal • ∫ 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} (c : ℝ≥0∞)
Goal: ∫ (x : ℝ) in a..b, f x ∂c • μ = c.toReal • ∫ (x : ℝ) in a..b, f x ∂μ
-- RCLike.intervalIntegral_ofReal
nonrec theorem _root_.RCLike.intervalIntegral_ofReal {𝕜 : Type*} [RCLike 𝕜] {a b : ℝ}
    {μ : Measure ℝ} {f : ℝ → ℝ} : (∫ x in a..b, (f x : 𝕜) ∂μ) = ↑(∫ x in a..b, f x ∂μ) 
Arguments: {𝕜 : Type u_6} [RCLike 𝕜] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → ℝ}
Goal: ∫ (x : ℝ) in a..b, ↑(f x) ∂μ = ↑(∫ (x : ℝ) in a..b, f x ∂μ)
-- intervalIntegral.integral_ofReal
nonrec theorem integral_ofReal {a b : ℝ} {μ : Measure ℝ} {f : ℝ → ℝ} :
    (∫ x in a..b, (f x : ℂ) ∂μ) = ↑(∫ x in 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 a..b, f x ∂μ)
-- ContinuousLinearMap.intervalIntegral_apply
theorem _root_.ContinuousLinearMap.intervalIntegral_apply {a b : ℝ} {φ : ℝ → F →L[𝕜] E}
    (hφ : IntervalIntegrable φ μ a b) (v : F) :
    (∫ x in a..b, φ x ∂μ) v = ∫ x in a..b, φ x v ∂μ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {a b : ℝ} {φ : ℝ → F →L[𝕜] E} (hφ : IntervalIntegrable φ μ a b) (v : F)
Goal: (∫ (x : ℝ) in a..b, φ x ∂μ) v = ∫ (x : ℝ) in a..b, (φ x) v ∂μ
-- ContinuousLinearMap.intervalIntegral_comp_comm
theorem _root_.ContinuousLinearMap.intervalIntegral_comp_comm (L : E →L[𝕜] F)
    (hf : IntervalIntegrable f μ a b) : (∫ x in a..b, L (f x) ∂μ) = L (∫ x in a..b, f x ∂μ) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} {F : Type u_4} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedSpace ℝ F] [CompleteSpace F] (L : E →L[𝕜] F) (hf : IntervalIntegrable f μ a b)
Goal: ∫ (x : ℝ) in a..b, L (f x) ∂μ = L (∫ (x : ℝ) in a..b, f x ∂μ)
-- intervalIntegral.integral_comp_mul_right
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_mul_right (hc : c ≠ 0) :
    (∫ x in a..b, f (x * c)) = c⁻¹ • ∫ x in a * c..b * c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0)
Goal: ∫ (x : ℝ) in a..b, f (x * c) = c⁻¹ • ∫ (x : ℝ) in a * c..b * c, f x
-- intervalIntegral.smul_integral_comp_mul_right
@[simp] -- Porting note (#10618): was @[simp]
theorem smul_integral_comp_mul_right (c) :
    (c • ∫ x in a..b, f (x * c)) = ∫ x in a * c..b * c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (x * c) = ∫ (x : ℝ) in a * c..b * c, f x
-- intervalIntegral.integral_comp_mul_left
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_mul_left (hc : c ≠ 0) :
    (∫ x in a..b, f (c * x)) = c⁻¹ • ∫ x in c * a..c * b, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0)
Goal: ∫ (x : ℝ) in a..b, f (c * x) = c⁻¹ • ∫ (x : ℝ) in c * a..c * b, f x
-- intervalIntegral.smul_integral_comp_mul_left
@[simp] -- Porting note (#10618): was @[simp]
theorem smul_integral_comp_mul_left (c) :
    (c • ∫ x in a..b, f (c * x)) = ∫ x in c * a..c * b, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (c * x) = ∫ (x : ℝ) in c * a..c * b, f x
-- intervalIntegral.integral_comp_div
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_div (hc : c ≠ 0) :
    (∫ x in a..b, f (x / c)) = c • ∫ x in a / c..b / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0)
Goal: ∫ (x : ℝ) in a..b, f (x / c) = c • ∫ (x : ℝ) in a / c..b / c, f x
-- intervalIntegral.inv_smul_integral_comp_div
@[simp] -- Porting note (#10618): was @[simp]
theorem inv_smul_integral_comp_div (c) :
    (c⁻¹ • ∫ x in a..b, f (x / c)) = ∫ x in a / c..b / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c : ℝ)
Goal: c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c) = ∫ (x : ℝ) in a / c..b / c, f x
-- intervalIntegral.integral_comp_add_right
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_add_right (d) : (∫ x in a..b, f (x + d)) = ∫ x in a + d..b + d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (x + d) = ∫ (x : ℝ) in a + d..b + d, f x
-- intervalIntegral.integral_comp_add_left
@[simp] -- Porting note (#10618): was @[simp]
nonrec theorem integral_comp_add_left (d) :
    (∫ x in a..b, f (d + x)) = ∫ x in d + a..d + b, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d + x) = ∫ (x : ℝ) in d + a..d + b, f x
-- intervalIntegral.integral_comp_mul_add
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_mul_add (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (c * x + d)) = c⁻¹ • ∫ x in c * a + d..c * b + d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (c * x + d) = c⁻¹ • ∫ (x : ℝ) in c * a + d..c * b + d, f x
-- intervalIntegral.smul_integral_comp_mul_add
@[simp] -- Porting note (#10618): was @[simp]
theorem smul_integral_comp_mul_add (c d) :
    (c • ∫ x in a..b, f (c * x + d)) = ∫ x in c * a + d..c * b + d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (c * x + d) = ∫ (x : ℝ) in c * a + d..c * b + d, f x
-- intervalIntegral.integral_comp_add_mul
@[simp] -- Porting note (#10618): was @[simp]
theorem integral_comp_add_mul (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (d + c * x)) = c⁻¹ • ∫ x in d + c * a..d + c * b, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d + c * x) = c⁻¹ • ∫ (x : ℝ) in d + c * a..d + c * b, f x
-- intervalIntegral.smul_integral_comp_add_mul
theorem smul_integral_comp_add_mul (c d) :
    (c • ∫ x in a..b, f (d + c * x)) = ∫ x in d + c * a..d + c * b, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (d + c * x) = ∫ (x : ℝ) in d + c * a..d + c * b, f x
-- intervalIntegral.integral_comp_div_add
theorem integral_comp_div_add (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (x / c + d)) = c • ∫ x in a / c + d..b / c + d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (x / c + d) = c • ∫ (x : ℝ) in a / c + d..b / c + d, f x
-- intervalIntegral.inv_smul_integral_comp_div_add
theorem inv_smul_integral_comp_div_add (c d) :
    (c⁻¹ • ∫ x in a..b, f (x / c + d)) = ∫ x in a / c + d..b / c + d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c + d) = ∫ (x : ℝ) in a / c + d..b / c + d, f x
-- intervalIntegral.integral_comp_add_div
theorem integral_comp_add_div (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (d + x / c)) = c • ∫ x in d + a / c..d + b / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d + x / c) = c • ∫ (x : ℝ) in d + a / c..d + b / c, f x
-- intervalIntegral.inv_smul_integral_comp_add_div
theorem inv_smul_integral_comp_add_div (c d) :
    (c⁻¹ • ∫ x in a..b, f (d + x / c)) = ∫ x in d + a / c..d + b / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c⁻¹ • ∫ (x : ℝ) in a..b, f (d + x / c) = ∫ (x : ℝ) in d + a / c..d + b / c, f x
-- intervalIntegral.integral_comp_mul_sub
theorem integral_comp_mul_sub (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (c * x - d)) = c⁻¹ • ∫ x in c * a - d..c * b - d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (c * x - d) = c⁻¹ • ∫ (x : ℝ) in c * a - d..c * b - d, f x
-- intervalIntegral.smul_integral_comp_mul_sub
theorem smul_integral_comp_mul_sub (c d) :
    (c • ∫ x in a..b, f (c * x - d)) = ∫ x in c * a - d..c * b - d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (c * x - d) = ∫ (x : ℝ) in c * a - d..c * b - d, f x
-- intervalIntegral.integral_comp_sub_mul
theorem integral_comp_sub_mul (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (d - c * x)) = c⁻¹ • ∫ x in d - c * b..d - c * a, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d - c * x) = c⁻¹ • ∫ (x : ℝ) in d - c * b..d - c * a, f x
-- intervalIntegral.smul_integral_comp_sub_mul
theorem smul_integral_comp_sub_mul (c d) :
    (c • ∫ x in a..b, f (d - c * x)) = ∫ x in d - c * b..d - c * a, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c • ∫ (x : ℝ) in a..b, f (d - c * x) = ∫ (x : ℝ) in d - c * b..d - c * a, f x
-- intervalIntegral.integral_comp_div_sub
theorem integral_comp_div_sub (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (x / c - d)) = c • ∫ x in a / c - d..b / c - d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (x / c - d) = c • ∫ (x : ℝ) in a / c - d..b / c - d, f x
-- intervalIntegral.inv_smul_integral_comp_div_sub
theorem inv_smul_integral_comp_div_sub (c d) :
    (c⁻¹ • ∫ x in a..b, f (x / c - d)) = ∫ x in a / c - d..b / c - d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c⁻¹ • ∫ (x : ℝ) in a..b, f (x / c - d) = ∫ (x : ℝ) in a / c - d..b / c - d, f x
-- intervalIntegral.integral_comp_sub_div
theorem integral_comp_sub_div (hc : c ≠ 0) (d) :
    (∫ x in a..b, f (d - x / c)) = c • ∫ x in d - b / c..d - a / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} (f : ℝ → E) (hc : c ≠ 0) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d - x / c) = c • ∫ (x : ℝ) in d - b / c..d - a / c, f x
-- intervalIntegral.inv_smul_integral_comp_sub_div
theorem inv_smul_integral_comp_sub_div (c d) :
    (c⁻¹ • ∫ x in a..b, f (d - x / c)) = ∫ x in d - b / c..d - a / c, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (c d : ℝ)
Goal: c⁻¹ • ∫ (x : ℝ) in a..b, f (d - x / c) = ∫ (x : ℝ) in d - b / c..d - a / c, f x
-- intervalIntegral.integral_comp_sub_right
theorem integral_comp_sub_right (d) : (∫ x in a..b, f (x - d)) = ∫ x in a - d..b - d, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (x - d) = ∫ (x : ℝ) in a - d..b - d, f x
-- intervalIntegral.integral_comp_sub_left
theorem integral_comp_sub_left (d) : (∫ x in a..b, f (d - x)) = ∫ x in d - b..d - a, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E) (d : ℝ)
Goal: ∫ (x : ℝ) in a..b, f (d - x) = ∫ (x : ℝ) in d - b..d - a, f x
-- intervalIntegral.integral_comp_neg
theorem integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} (f : ℝ → E)
Goal: ∫ (x : ℝ) in a..b, f (-x) = ∫ (x : ℝ) in -b..-a, f x
-- intervalIntegral.integral_congr
theorem integral_congr {a b : ℝ} (h : EqOn f g [[a, b]]) :
    ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a b : ℝ} (h : Set.EqOn f g [[a, b]])
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_add_adjacent_intervals_cancel
theorem integral_add_adjacent_intervals_cancel (hab : IntervalIntegrable f μ a b)
    (hbc : IntervalIntegrable f μ b c) :
    (((∫ x in a..b, f x ∂μ) + ∫ x in b..c, f x ∂μ) + ∫ x in c..a, f x ∂μ) = 0 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in b..c, f x ∂μ + ∫ (x : ℝ) in c..a, f x ∂μ = 0
-- intervalIntegral.integral_add_adjacent_intervals
theorem integral_add_adjacent_intervals (hab : IntervalIntegrable f μ a b)
    (hbc : IntervalIntegrable f μ b c) :
    ((∫ x in a..b, f x ∂μ) + ∫ x in b..c, f x ∂μ) = ∫ x in a..c, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in b..c, f x ∂μ = ∫ (x : ℝ) in a..c, f x ∂μ
-- intervalIntegral.sum_integral_adjacent_intervals_Ico
theorem sum_integral_adjacent_intervals_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
    (hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
    ∑ k ∈ Finset.Ico m n, ∫ x in a k..a <| k + 1, f x ∂μ = ∫ x in a m..a n, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n) (hint : ∀ k ∈ Set.Ico m n, IntervalIntegrable f μ (a k) (a (k + 1)))
Goal: ∑ k ∈ Finset.Ico m n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a m..a n, f x ∂μ
-- intervalIntegral.sum_integral_adjacent_intervals
theorem sum_integral_adjacent_intervals {a : ℕ → ℝ} {n : ℕ}
    (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
    ∑ k ∈ Finset.range n, ∫ x in a k..a <| k + 1, f x ∂μ = ∫ x in (a 0)..(a n), f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a (k + 1)))
Goal: ∑ k ∈ Finset.range n, ∫ (x : ℝ) in a k..a (k + 1), f x ∂μ = ∫ (x : ℝ) in a 0 ..a n, f x ∂μ
-- intervalIntegral.integral_interval_sub_left
theorem integral_interval_sub_left (hab : IntervalIntegrable f μ a b)
    (hac : IntervalIntegrable f μ a c) :
    ((∫ x in a..b, f x ∂μ) - ∫ x in a..c, f x ∂μ) = ∫ x in c..b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hac : IntervalIntegrable f μ a c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in a..c, f x ∂μ = ∫ (x : ℝ) in c..b, f x ∂μ
-- intervalIntegral.integral_interval_add_interval_comm
theorem integral_interval_add_interval_comm (hab : IntervalIntegrable f μ a b)
    (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) :
    ((∫ x in a..b, f x ∂μ) + ∫ x in c..d, f x ∂μ) =
      (∫ x in a..d, f x ∂μ) + ∫ x in c..b, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c d : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..d, f x ∂μ + ∫ (x : ℝ) in c..b, f x ∂μ
-- intervalIntegral.integral_interval_sub_interval_comm
theorem integral_interval_sub_interval_comm (hab : IntervalIntegrable f μ a b)
    (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) :
    ((∫ x in a..b, f x ∂μ) - ∫ x in c..d, f x ∂μ) =
      (∫ x in a..c, f x ∂μ) - ∫ x in b..d, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c d : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in a..c, f x ∂μ - ∫ (x : ℝ) in b..d, f x ∂μ
-- intervalIntegral.integral_interval_sub_interval_comm'
theorem integral_interval_sub_interval_comm' (hab : IntervalIntegrable f μ a b)
    (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c) :
    ((∫ x in a..b, f x ∂μ) - ∫ x in c..d, f x ∂μ) =
      (∫ x in d..b, f x ∂μ) - ∫ x in c..a, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b c d : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (hab : IntervalIntegrable f μ a b) (hcd : IntervalIntegrable f μ c d) (hac : IntervalIntegrable f μ a c)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in d..b, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ
-- intervalIntegral.integral_Iic_sub_Iic
theorem integral_Iic_sub_Iic (ha : IntegrableOn f (Iic a) μ) (hb : IntegrableOn f (Iic b) μ) :
    ((∫ x in Iic b, f x ∂μ) - ∫ x in Iic a, 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} (ha : MeasureTheory.IntegrableOn f (Set.Iic a) μ) (hb : MeasureTheory.IntegrableOn f (Set.Iic b) μ)
Goal: ∫ (x : ℝ) in Set.Iic b, f x ∂μ - ∫ (x : ℝ) in Set.Iic a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
-- intervalIntegral.integral_Iic_add_Ioi
theorem integral_Iic_add_Ioi (h_left : IntegrableOn f (Iic b) μ)
    (h_right : IntegrableOn f (Ioi b) μ) :
    (∫ x in Iic b, f x ∂μ) + (∫ x in Ioi b, f x ∂μ) = ∫ (x : ℝ), f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h_left : MeasureTheory.IntegrableOn f (Set.Iic b) μ) (h_right : MeasureTheory.IntegrableOn f (Set.Ioi b) μ)
Goal: ∫ (x : ℝ) in Set.Iic b, f x ∂μ + ∫ (x : ℝ) in Set.Ioi b, f x ∂μ = ∫ (x : ℝ), f x ∂μ
-- intervalIntegral.integral_Iio_add_Ici
theorem integral_Iio_add_Ici (h_left : IntegrableOn f (Iio b) μ)
    (h_right : IntegrableOn f (Ici b) μ) :
    (∫ x in Iio b, f x ∂μ) + (∫ x in Ici b, f x ∂μ) = ∫ (x : ℝ), f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ℝ} {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} (h_left : MeasureTheory.IntegrableOn f (Set.Iio b) μ) (h_right : MeasureTheory.IntegrableOn f (Set.Ici b) μ)
Goal: ∫ (x : ℝ) in Set.Iio b, f x ∂μ + ∫ (x : ℝ) in Set.Ici b, f x ∂μ = ∫ (x : ℝ), f x ∂μ
-- intervalIntegral.integral_const_of_cdf
theorem integral_const_of_cdf [IsFiniteMeasure μ] (c : E) :
    ∫ _ in a..b, c ∂μ = ((μ (Iic b)).toReal - (μ (Iic a)).toReal) • c 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsFiniteMeasure μ] (c : E)
Goal: ∫ (x : ℝ) in a..b, c ∂μ = ((μ (Set.Iic b)).toReal - (μ (Set.Iic a)).toReal) • c
-- intervalIntegral.integral_eq_integral_of_support_subset
theorem integral_eq_integral_of_support_subset {a b} (h : support f ⊆ Ioc a b) :
    ∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a b : ℝ} (h : Function.support f ⊆ Set.Ioc a b)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ), f x ∂μ
-- intervalIntegral.integral_congr_ae'
theorem integral_congr_ae' (h : ∀ᵐ x ∂μ, x ∈ Ioc a b → f x = g x)
    (h' : ∀ᵐ x ∂μ, x ∈ Ioc b a → f x = g x) : ∫ 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} (h : ∀ᵐ (x : ℝ) ∂μ, x ∈ Set.Ioc a b → f x = g x) (h' : ∀ᵐ (x : ℝ) ∂μ, x ∈ Set.Ioc b a → f x = g x)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_congr_ae
theorem integral_congr_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = g x) :
    ∫ 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} (h : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → f x = g x)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ
-- intervalIntegral.integral_zero_ae
theorem integral_zero_ae (h : ∀ᵐ x ∂μ, x ∈ Ι a b → f x = 0) : ∫ 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 : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → f x = 0)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0
-- intervalIntegral.integral_indicator
nonrec theorem integral_indicator {a₁ a₂ a₃ : ℝ} (h : a₂ ∈ Icc a₁ a₃) :
    ∫ x in a₁..a₃, indicator {x | x ≤ a₂} f x ∂μ = ∫ x in a₁..a₂, f x ∂μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a₁ a₂ a₃ : ℝ} (h : a₂ ∈ Set.Icc a₁ a₃)
Goal: ∫ (x : ℝ) in a₁..a₃, {x | x ≤ a₂}.indicator f x ∂μ = ∫ (x : ℝ) in a₁..a₂, f x ∂μ
-- intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae
theorem integral_eq_zero_iff_of_le_of_nonneg_ae (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Ioc a b)] f)
    (hfi : IntervalIntegrable f μ a b) :
    ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b)] 0 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Set.Ioc a b)] f) (hfi : IntervalIntegrable f μ a b)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Set.Ioc a b)] 0
-- intervalIntegral.integral_eq_zero_iff_of_nonneg_ae
theorem integral_eq_zero_iff_of_nonneg_ae (hf : 0 ≤ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] f)
    (hfi : IntervalIntegrable f μ a b) :
    ∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b ∪ Ioc b a)] 0 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : 0 ≤ᵐ[μ.restrict (Set.Ioc a b ∪ Set.Ioc b a)] f) (hfi : IntervalIntegrable f μ a b)
Goal: ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Set.Ioc a b ∪ Set.Ioc b a)] 0
-- intervalIntegral.integral_pos_iff_support_of_nonneg_ae'
theorem integral_pos_iff_support_of_nonneg_ae' (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f)
    (hfi : IntervalIntegrable f μ a b) :
    (0 < ∫ x in a..b, f x ∂μ) ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : 0 ≤ᵐ[μ.restrict (Ι a b)] f) (hfi : IntervalIntegrable f μ a b)
Goal: 0 < ∫ (x : ℝ) in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (Function.support f ∩ Set.Ioc a b)
-- intervalIntegral.integral_pos_iff_support_of_nonneg_ae
theorem integral_pos_iff_support_of_nonneg_ae (hf : 0 ≤ᵐ[μ] f) (hfi : IntervalIntegrable f μ a b) :
    (0 < ∫ x in a..b, f x ∂μ) ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) 
Arguments: {f : ℝ → ℝ} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : IntervalIntegrable f μ a b)
Goal: 0 < ∫ (x : ℝ) in a..b, f x ∂μ ↔ a < b ∧ 0 < μ (Function.support f ∩ Set.Ioc a b)