ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le'
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le' [IsMeasurablyGenerated l']
    [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
    (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
    (hv : Tendsto v lt l) (huv : u ≤ᶠ[lt] v) :
    (fun t => (∫ x in u t..v t, f x ∂μ) - (μ (Ioc (u t) (v t))).toReal • c) =o[lt] fun t =>
      (μ <| Ioc (u t) (v t)).toReal 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [CompleteSpace E] [l'.IsMeasurablyGenerated] [Filter.TendstoIxxClass Set.Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l) (huv : u ≤ᶠ[lt] v)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - (μ (Set.Ioc (u t) (v t))).toReal • c) =o[lt] fun t =>
  (μ (Set.Ioc (u t) (v t))).toReal
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge'
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge' [IsMeasurablyGenerated l']
    [TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
    (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
    (hv : Tendsto v lt l) (huv : v ≤ᶠ[lt] u) :
    (fun t => (∫ x in u t..v t, f x ∂μ) + (μ (Ioc (v t) (u t))).toReal • c) =o[lt] fun t =>
      (μ <| Ioc (v t) (u t)).toReal 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [CompleteSpace E] [l'.IsMeasurablyGenerated] [Filter.TendstoIxxClass Set.Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l) (huv : v ≤ᶠ[lt] u)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ + (μ (Set.Ioc (v t) (u t))).toReal • c) =o[lt] fun t =>
  (μ (Set.Ioc (v t) (u t))).toReal
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae
    (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c))
    (hu : Tendsto u lt l) (hv : Tendsto v lt l) :
    (fun t => (∫ x in u t..v t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t =>
      ∫ _ in u t..v t, (1 : ℝ) ∂μ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a : ℝ} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] [intervalIntegral.FTCFilter a l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_le
    (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c))
    (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : u ≤ᶠ[lt] v) :
    (fun t => (∫ x in u t..v t, f x ∂μ) - (μ (Ioc (u t) (v t))).toReal • c) =o[lt] fun t =>
      (μ <| Ioc (u t) (v t)).toReal 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a : ℝ} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [CompleteSpace E] [MeasureTheory.IsLocallyFiniteMeasure μ] [intervalIntegral.FTCFilter a l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l) (huv : u ≤ᶠ[lt] v)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - (μ (Set.Ioc (u t) (v t))).toReal • c) =o[lt] fun t =>
  (μ (Set.Ioc (u t) (v t))).toReal
-- intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae_of_ge
    (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Tendsto f (l' ⊓ ae μ) (𝓝 c))
    (hu : Tendsto u lt l) (hv : Tendsto v lt l) (huv : v ≤ᶠ[lt] u) :
    (fun t => (∫ x in u t..v t, f x ∂μ) + (μ (Ioc (v t) (u t))).toReal • c) =o[lt] fun t =>
      (μ <| Ioc (v t) (u t)).toReal 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a : ℝ} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [CompleteSpace E] [MeasureTheory.IsLocallyFiniteMeasure μ] [intervalIntegral.FTCFilter a l l'] (hfm : StronglyMeasurableAtFilter f l' μ) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l) (huv : v ≤ᶠ[lt] u)
Goal: (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ + (μ (Set.Ioc (v t) (u t))).toReal • c) =o[lt] fun t =>
  (μ (Set.Ioc (v t) (u t))).toReal
-- intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae
theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae
    (hab : IntervalIntegrable f μ a b) (hmeas_a : StronglyMeasurableAtFilter f la' μ)
    (hmeas_b : StronglyMeasurableAtFilter f lb' μ) (ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca))
    (hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)) (hua : Tendsto ua lt la) (hva : Tendsto va lt la)
    (hub : Tendsto ub lt lb) (hvb : Tendsto vb lt lb) :
    (fun t =>
        ((∫ x in va t..vb t, f x ∂μ) - ∫ x in ua t..ub t, f x ∂μ) -
          ((∫ _ in ub t..vb t, cb ∂μ) - ∫ _ in ua t..va t, ca ∂μ)) =o[lt]
      fun t => ‖∫ _ in ua t..va t, (1 : ℝ) ∂μ‖ + ‖∫ _ in ub t..vb t, (1 : ℝ) ∂μ‖ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {ca cb : E} {la la' lb lb' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {ua va ub vb : ι → ℝ} [intervalIntegral.FTCFilter a la la'] [intervalIntegral.FTCFilter b lb lb'] [MeasureTheory.IsLocallyFiniteMeasure μ] (hab : IntervalIntegrable f μ a b) (hmeas_a : StronglyMeasurableAtFilter f la' μ) (hmeas_b : StronglyMeasurableAtFilter f lb' μ) (ha_lim : Filter.Tendsto f (la' ⊓ MeasureTheory.ae μ) (𝓝 ca)) (hb_lim : Filter.Tendsto f (lb' ⊓ MeasureTheory.ae μ) (𝓝 cb)) (hua : Filter.Tendsto ua lt la) (hva : Filter.Tendsto va lt la) (hub : Filter.Tendsto ub lt lb) (hvb : Filter.Tendsto vb lt lb)
Goal: (fun t =>
    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -
      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]
  fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖
-- intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right
theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right
    (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f lb' μ)
    (hf : Tendsto f (lb' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt lb) (hv : Tendsto v lt lb) :
    (fun t => ((∫ x in a..v t, f x ∂μ) - ∫ x in a..u t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt]
      fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {c : E} {lb lb' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [intervalIntegral.FTCFilter b lb lb'] [MeasureTheory.IsLocallyFiniteMeasure μ] (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f lb' μ) (hf : Filter.Tendsto f (lb' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hu : Filter.Tendsto u lt lb) (hv : Filter.Tendsto v lt lb)
Goal: (fun t => ∫ (x : ℝ) in a..v t, f x ∂μ - ∫ (x : ℝ) in a..u t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t =>
  ∫ (x : ℝ) in u t..v t, 1 ∂μ
-- intervalIntegral.measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left
theorem measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left
    (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f la' μ)
    (hf : Tendsto f (la' ⊓ ae μ) (𝓝 c)) (hu : Tendsto u lt la) (hv : Tendsto v lt la) :
    (fun t => ((∫ x in v t..b, f x ∂μ) - ∫ x in u t..b, f x ∂μ) + ∫ _ in u t..v t, c ∂μ) =o[lt]
      fun t => ∫ _ in u t..v t, (1 : ℝ) ∂μ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} {c : E} {la la' : Filter ℝ} {lt : Filter ι} {μ : MeasureTheory.Measure ℝ} {u v : ι → ℝ} [intervalIntegral.FTCFilter a la la'] [MeasureTheory.IsLocallyFiniteMeasure μ] (hab : IntervalIntegrable f μ a b) (hmeas : StronglyMeasurableAtFilter f la' μ) (hf : Filter.Tendsto f (la' ⊓ MeasureTheory.ae μ) (𝓝 c)) (hu : Filter.Tendsto u lt la) (hv : Filter.Tendsto v lt la)
Goal: (fun t => ∫ (x : ℝ) in v t..b, f x ∂μ - ∫ (x : ℝ) in u t..b, f x ∂μ + ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t =>
  ∫ (x : ℝ) in u t..v t, 1 ∂μ
-- intervalIntegral.integral_sub_linear_isLittleO_of_tendsto_ae
theorem integral_sub_linear_isLittleO_of_tendsto_ae [FTCFilter a l l']
    (hfm : StronglyMeasurableAtFilter f l') (hf : Tendsto f (l' ⊓ ae volume) (𝓝 c)) {u v : ι → ℝ}
    (hu : Tendsto u lt l) (hv : Tendsto v lt l) :
    (fun t => (∫ x in u t..v t, f x) - (v t - u t) • c) =o[lt] (v - u) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {l l' : Filter ℝ} {lt : Filter ι} {a : ℝ} [intervalIntegral.FTCFilter a l l'] (hfm : StronglyMeasurableAtFilter f l' MeasureTheory.volume) (hf : Filter.Tendsto f (l' ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)) {u v : ι → ℝ} (hu : Filter.Tendsto u lt l) (hv : Filter.Tendsto v lt l)
Goal: (fun t => (∫ (x : ℝ) in u t..v t, f x) - (v t - u t) • c) =o[lt] (v - u)
-- intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae
theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae
    (hab : IntervalIntegrable f volume a b) (hmeas_a : StronglyMeasurableAtFilter f la')
    (hmeas_b : StronglyMeasurableAtFilter f lb') (ha_lim : Tendsto f (la' ⊓ ae volume) (𝓝 ca))
    (hb_lim : Tendsto f (lb' ⊓ ae volume) (𝓝 cb)) (hua : Tendsto ua lt la) (hva : Tendsto va lt la)
    (hub : Tendsto ub lt lb) (hvb : Tendsto vb lt lb) :
    (fun t =>
        ((∫ x in va t..vb t, f x) - ∫ x in ua t..ub t, f x) -
          ((vb t - ub t) • cb - (va t - ua t) • ca)) =o[lt]
      fun t => ‖va t - ua t‖ + ‖vb t - ub t‖ 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : E} {la la' lb lb' : Filter ℝ} {lt : Filter ι} {a b : ℝ} {ua ub va vb : ι → ℝ} [intervalIntegral.FTCFilter a la la'] [intervalIntegral.FTCFilter b lb lb'] (hab : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f la' MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f lb' MeasureTheory.volume) (ha_lim : Filter.Tendsto f (la' ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb_lim : Filter.Tendsto f (lb' ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb)) (hua : Filter.Tendsto ua lt la) (hva : Filter.Tendsto va lt la) (hub : Filter.Tendsto ub lt lb) (hvb : Filter.Tendsto vb lt lb)
Goal: (fun t =>
    ((∫ (x : ℝ) in va t..vb t, f x) - ∫ (x : ℝ) in ua t..ub t, f x) - ((vb t - ub t) • cb - (va t - ua t) • ca)) =o[lt]
  fun t => ‖va t - ua t‖ + ‖vb t - ub t‖
-- intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right
theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_right
    (hab : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f lb')
    (hf : Tendsto f (lb' ⊓ ae volume) (𝓝 c)) (hu : Tendsto u lt lb) (hv : Tendsto v lt lb) :
    (fun t => ((∫ x in a..v t, f x) - ∫ x in a..u t, f x) - (v t - u t) • c) =o[lt] (v - u) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {lb lb' : Filter ℝ} {lt : Filter ι} {a b : ℝ} {u v : ι → ℝ} [intervalIntegral.FTCFilter b lb lb'] (hab : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f lb' MeasureTheory.volume) (hf : Filter.Tendsto f (lb' ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)) (hu : Filter.Tendsto u lt lb) (hv : Filter.Tendsto v lt lb)
Goal: (fun t => ((∫ (x : ℝ) in a..v t, f x) - ∫ (x : ℝ) in a..u t, f x) - (v t - u t) • c) =o[lt] (v - u)
-- intervalIntegral.integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left
theorem integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left
    (hab : IntervalIntegrable f volume a b) (hmeas : StronglyMeasurableAtFilter f la')
    (hf : Tendsto f (la' ⊓ ae volume) (𝓝 c)) (hu : Tendsto u lt la) (hv : Tendsto v lt la) :
    (fun t => ((∫ x in v t..b, f x) - ∫ x in u t..b, f x) + (v t - u t) • c) =o[lt] (v - u) 
Arguments: {ι : Type u_1} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {la la' : Filter ℝ} {lt : Filter ι} {a b : ℝ} {u v : ι → ℝ} [intervalIntegral.FTCFilter a la la'] (hab : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f la' MeasureTheory.volume) (hf : Filter.Tendsto f (la' ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)) (hu : Filter.Tendsto u lt la) (hv : Filter.Tendsto v lt la)
Goal: (fun t => ((∫ (x : ℝ) in v t..b, f x) - ∫ (x : ℝ) in u t..b, f x) + (v t - u t) • c) =o[lt] (v - u)
-- intervalIntegral.integral_hasStrictFDerivAt_of_tendsto_ae
theorem integral_hasStrictFDerivAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) :
    HasStrictFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x)
      ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb : Filter.Tendsto f (𝓝 b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb))
Goal: HasStrictFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x)
  ((ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight cb - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight ca) (a, b)
-- intervalIntegral.integral_hasStrictFDerivAt
theorem integral_hasStrictFDerivAt (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : ContinuousAt f a) (hb : ContinuousAt f b) :
    HasStrictFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x)
      ((snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a)) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : ContinuousAt f a) (hb : ContinuousAt f b)
Goal: HasStrictFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x)
  ((ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight (f b) - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight (f a)) (a, b)
-- intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_right
theorem integral_hasStrictDerivAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) :
    HasStrictDerivAt (fun u => ∫ x in a..u, f x) c b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (hb : Filter.Tendsto f (𝓝 b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: HasStrictDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) c b
-- intervalIntegral.integral_hasStrictDerivAt_right
theorem integral_hasStrictDerivAt_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) :
    HasStrictDerivAt (fun u => ∫ x in a..u, f x) (f b) b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (hb : ContinuousAt f b)
Goal: HasStrictDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) b
-- intervalIntegral.integral_hasStrictDerivAt_of_tendsto_ae_left
theorem integral_hasStrictDerivAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) :
    HasStrictDerivAt (fun u => ∫ x in u..b, f x) (-c) a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: HasStrictDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) a
-- intervalIntegral.integral_hasStrictDerivAt_left
theorem integral_hasStrictDerivAt_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : ContinuousAt f a) :
    HasStrictDerivAt (fun u => ∫ x in u..b, f x) (-f a) a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (ha : ContinuousAt f a)
Goal: HasStrictDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) a
-- Continuous.integral_hasStrictDerivAt
theorem _root_.Continuous.integral_hasStrictDerivAt {f : ℝ → E} (hf : Continuous f) (a b : ℝ) :
    HasStrictDerivAt (fun u => ∫ x : ℝ in a..u, f x) (f b) b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} (hf : Continuous f) (a b : ℝ)
Goal: HasStrictDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) b
-- Continuous.deriv_integral
theorem _root_.Continuous.deriv_integral (f : ℝ → E) (hf : Continuous f) (a b : ℝ) :
    deriv (fun u => ∫ x : ℝ in a..u, f x) b = f b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f : ℝ → E) (hf : Continuous f) (a b : ℝ)
Goal: deriv (fun u => ∫ (x : ℝ) in a..u, f x) b = f b
-- intervalIntegral.integral_hasFDerivAt_of_tendsto_ae
theorem integral_hasFDerivAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) :
    HasFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x)
      ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb : Filter.Tendsto f (𝓝 b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb))
Goal: HasFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x)
  ((ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight cb - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight ca) (a, b)
-- intervalIntegral.integral_hasFDerivAt
theorem integral_hasFDerivAt (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : ContinuousAt f a) (hb : ContinuousAt f b) :
    HasFDerivAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x)
      ((snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a)) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : ContinuousAt f a) (hb : ContinuousAt f b)
Goal: HasFDerivAt (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x)
  ((ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight (f b) - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight (f a)) (a, b)
-- intervalIntegral.fderiv_integral_of_tendsto_ae
theorem fderiv_integral_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 cb)) :
    fderiv ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (a, b) =
      (snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb : Filter.Tendsto f (𝓝 b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb))
Goal: fderiv ℝ (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (a, b) =
  (ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight cb - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight ca
-- intervalIntegral.fderiv_integral
theorem fderiv_integral (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a)) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b))
    (ha : ContinuousAt f a) (hb : ContinuousAt f b) :
    fderiv ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (a, b) =
      (snd ℝ ℝ ℝ).smulRight (f b) - (fst ℝ ℝ ℝ).smulRight (f a) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (ha : ContinuousAt f a) (hb : ContinuousAt f b)
Goal: fderiv ℝ (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (a, b) =
  (ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight (f b) - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight (f a)
-- intervalIntegral.integral_hasDerivAt_of_tendsto_ae_right
theorem integral_hasDerivAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) :
    HasDerivAt (fun u => ∫ x in a..u, f x) c b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {c : E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (hb : Filter.Tendsto f (𝓝 b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: HasDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) c b
-- intervalIntegral.integral_hasDerivAt_right
theorem integral_hasDerivAt_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) :
    HasDerivAt (fun u => ∫ x in a..u, f x) (f b) b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas : StronglyMeasurableAtFilter f (𝓝 b) MeasureTheory.volume) (hb : ContinuousAt f b)
Goal: HasDerivAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) b