ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.deriv_integral_of_tendsto_ae_right
theorem deriv_integral_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : Tendsto f (𝓝 b ⊓ ae volume) (𝓝 c)) :
    deriv (fun u => ∫ x in a..u, f x) b = c 
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: deriv (fun u => ∫ (x : ℝ) in a..u, f x) b = c
-- intervalIntegral.deriv_integral_right
theorem deriv_integral_right (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 b)) (hb : ContinuousAt f b) :
    deriv (fun u => ∫ x in a..u, f x) b = f 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: deriv (fun u => ∫ (x : ℝ) in a..u, f x) b = f b
-- intervalIntegral.integral_hasDerivAt_of_tendsto_ae_left
theorem integral_hasDerivAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) :
    HasDerivAt (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: HasDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) a
-- intervalIntegral.integral_hasDerivAt_left
theorem integral_hasDerivAt_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (ha : ContinuousAt f a) :
    HasDerivAt (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: HasDerivAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) a
-- intervalIntegral.deriv_integral_of_tendsto_ae_left
theorem deriv_integral_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (hb : Tendsto f (𝓝 a ⊓ ae volume) (𝓝 c)) :
    deriv (fun u => ∫ x in u..b, f x) a = -c 
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) (hb : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: deriv (fun u => ∫ (x : ℝ) in u..b, f x) a = -c
-- intervalIntegral.deriv_integral_left
theorem deriv_integral_left (hf : IntervalIntegrable f volume a b)
    (hmeas : StronglyMeasurableAtFilter f (𝓝 a)) (hb : ContinuousAt f a) :
    deriv (fun u => ∫ x in u..b, f x) a = -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 : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume) (hb : ContinuousAt f a)
Goal: deriv (fun u => ∫ (x : ℝ) in u..b, f x) a = -f a
-- intervalIntegral.integral_hasFDerivWithinAt_of_tendsto_ae
theorem integral_hasFDerivWithinAt_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
    {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb]
    (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb)
    (ha : Tendsto f (la ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (lb ⊓ ae volume) (𝓝 cb)) :
    HasFDerivWithinAt (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x)
      ((snd ℝ ℝ ℝ).smulRight cb - (fst ℝ ℝ ℝ).smulRight ca) (s ×ˢ t) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {ca cb : E} {la lb : Filter ℝ} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) la] [intervalIntegral.FTCFilter b (𝓝[t] b) lb] (hmeas_a : StronglyMeasurableAtFilter f la MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f lb MeasureTheory.volume) (ha : Filter.Tendsto f (la ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb : Filter.Tendsto f (lb ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb))
Goal: HasFDerivWithinAt (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) (s ×ˢ t) (a, b)
-- intervalIntegral.integral_hasFDerivWithinAt
theorem integral_hasFDerivWithinAt (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb)
    {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb] (ha : Tendsto f la (𝓝 <| f a))
    (hb : Tendsto f lb (𝓝 <| f b)) :
    HasFDerivWithinAt (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)) (s ×ˢ t) (a, b) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {la lb : Filter ℝ} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f la MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f lb MeasureTheory.volume) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) la] [intervalIntegral.FTCFilter b (𝓝[t] b) lb] (ha : Filter.Tendsto f la (𝓝 (f a))) (hb : Filter.Tendsto f lb (𝓝 (f b)))
Goal: HasFDerivWithinAt (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)) (s ×ˢ t) (a, b)
-- intervalIntegral.fderivWithin_integral_of_tendsto_ae
theorem fderivWithin_integral_of_tendsto_ae (hf : IntervalIntegrable f volume a b)
    (hmeas_a : StronglyMeasurableAtFilter f la) (hmeas_b : StronglyMeasurableAtFilter f lb)
    {s t : Set ℝ} [FTCFilter a (𝓝[s] a) la] [FTCFilter b (𝓝[t] b) lb]
    (ha : Tendsto f (la ⊓ ae volume) (𝓝 ca)) (hb : Tendsto f (lb ⊓ ae volume) (𝓝 cb))
    (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ)
    (ht : UniqueDiffWithinAt ℝ t b := by uniqueDiffWithinAt_Ici_Iic_univ) :
    fderivWithin ℝ (fun p : ℝ × ℝ => ∫ x in p.1..p.2, f x) (s ×ˢ t) (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} {la lb : Filter ℝ} {a b : ℝ} (hf : IntervalIntegrable f MeasureTheory.volume a b) (hmeas_a : StronglyMeasurableAtFilter f la MeasureTheory.volume) (hmeas_b : StronglyMeasurableAtFilter f lb MeasureTheory.volume) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) la] [intervalIntegral.FTCFilter b (𝓝[t] b) lb] (ha : Filter.Tendsto f (la ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 ca)) (hb : Filter.Tendsto f (lb ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 cb)) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) (ht : UniqueDiffWithinAt ℝ t b := by uniqueDiffWithinAt_Ici_Iic_univ)
Goal: fderivWithin ℝ (fun p => ∫ (x : ℝ) in p.1 ..p.2, f x) (s ×ˢ t) (a, b) =
  (ContinuousLinearMap.snd ℝ ℝ ℝ).smulRight cb - (ContinuousLinearMap.fst ℝ ℝ ℝ).smulRight ca
-- intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_right
theorem integral_hasDerivWithinAt_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
    {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b))
    (hb : Tendsto f (𝓝[t] b ⊓ ae volume) (𝓝 c)) :
    HasDerivWithinAt (fun u => ∫ x in a..u, f x) c s 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) {s t : Set ℝ} [intervalIntegral.FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b) MeasureTheory.volume) (hb : Filter.Tendsto f (𝓝[t] b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: HasDerivWithinAt (fun u => ∫ (x : ℝ) in a..u, f x) c s b
-- intervalIntegral.integral_hasDerivWithinAt_right
theorem integral_hasDerivWithinAt_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ}
    [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b))
    (hb : ContinuousWithinAt f t b) : HasDerivWithinAt (fun u => ∫ x in a..u, f x) (f b) s 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) {s t : Set ℝ} [intervalIntegral.FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b) MeasureTheory.volume) (hb : ContinuousWithinAt f t b)
Goal: HasDerivWithinAt (fun u => ∫ (x : ℝ) in a..u, f x) (f b) s b
-- intervalIntegral.derivWithin_integral_of_tendsto_ae_right
theorem derivWithin_integral_of_tendsto_ae_right (hf : IntervalIntegrable f volume a b)
    {s t : Set ℝ} [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b))
    (hb : Tendsto f (𝓝[t] b ⊓ ae volume) (𝓝 c))
    (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ) :
    derivWithin (fun u => ∫ x in a..u, f x) s b = c 
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) {s t : Set ℝ} [intervalIntegral.FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b) MeasureTheory.volume) (hb : Filter.Tendsto f (𝓝[t] b ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)) (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ)
Goal: derivWithin (fun u => ∫ (x : ℝ) in a..u, f x) s b = c
-- intervalIntegral.derivWithin_integral_right
theorem derivWithin_integral_right (hf : IntervalIntegrable f volume a b) {s t : Set ℝ}
    [FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b))
    (hb : ContinuousWithinAt f t b)
    (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ) :
    derivWithin (fun u => ∫ x in a..u, f x) s b = f 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) {s t : Set ℝ} [intervalIntegral.FTCFilter b (𝓝[s] b) (𝓝[t] b)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] b) MeasureTheory.volume) (hb : ContinuousWithinAt f t b) (hs : UniqueDiffWithinAt ℝ s b := by uniqueDiffWithinAt_Ici_Iic_univ)
Goal: derivWithin (fun u => ∫ (x : ℝ) in a..u, f x) s b = f b
-- intervalIntegral.integral_hasDerivWithinAt_of_tendsto_ae_left
theorem integral_hasDerivWithinAt_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b)
    {s t : Set ℝ} [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a))
    (ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c)) :
    HasDerivWithinAt (fun u => ∫ x in u..b, f x) (-c) s 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) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝[t] a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c))
Goal: HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-c) s a
-- intervalIntegral.integral_hasDerivWithinAt_left
theorem integral_hasDerivWithinAt_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ}
    [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a))
    (ha : ContinuousWithinAt f t a) : HasDerivWithinAt (fun u => ∫ x in u..b, f x) (-f a) s 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) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) MeasureTheory.volume) (ha : ContinuousWithinAt f t a)
Goal: HasDerivWithinAt (fun u => ∫ (x : ℝ) in u..b, f x) (-f a) s a
-- intervalIntegral.derivWithin_integral_of_tendsto_ae_left
theorem derivWithin_integral_of_tendsto_ae_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ}
    [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a))
    (ha : Tendsto f (𝓝[t] a ⊓ ae volume) (𝓝 c))
    (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) :
    derivWithin (fun u => ∫ x in u..b, f x) s a = -c 
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) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) MeasureTheory.volume) (ha : Filter.Tendsto f (𝓝[t] a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ)
Goal: derivWithin (fun u => ∫ (x : ℝ) in u..b, f x) s a = -c
-- intervalIntegral.derivWithin_integral_left
theorem derivWithin_integral_left (hf : IntervalIntegrable f volume a b) {s t : Set ℝ}
    [FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a))
    (ha : ContinuousWithinAt f t a)
    (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ) :
    derivWithin (fun u => ∫ x in u..b, f x) s a = -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) {s t : Set ℝ} [intervalIntegral.FTCFilter a (𝓝[s] a) (𝓝[t] a)] (hmeas : StronglyMeasurableAtFilter f (𝓝[t] a) MeasureTheory.volume) (ha : ContinuousWithinAt f t a) (hs : UniqueDiffWithinAt ℝ s a := by uniqueDiffWithinAt_Ici_Iic_univ)
Goal: derivWithin (fun u => ∫ (x : ℝ) in u..b, f x) s a = -f a
-- intervalIntegral.differentiableOn_integral_of_continuous
theorem differentiableOn_integral_of_continuous {s : Set ℝ}
    (hintg : ∀ x ∈ s, IntervalIntegrable f volume a x) (hcont : Continuous f) :
    DifferentiableOn ℝ (fun u => ∫ x in a..u, f x) s 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : ℝ → E} {a : ℝ} {s : Set ℝ} (hintg : ∀ x ∈ s, IntervalIntegrable f MeasureTheory.volume a x) (hcont : Continuous f)
Goal: DifferentiableOn ℝ (fun u => ∫ (x : ℝ) in a..u, f x) s
-- intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le_Ico
theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a ≤ b)
    (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x)
    (φint : IntegrableOn φ (Icc a b)) (hφg : ∀ x ∈ Ico a b, g' x ≤ φ x) :
    g b - g a ≤ ∫ y in a..b, φ y 
Arguments: {g' g φ : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ico a b, HasDerivWithinAt g (g' x) (Set.Ioi x) x) (φint : MeasureTheory.IntegrableOn φ (Set.Icc a b) MeasureTheory.volume) (hφg : ∀ x ∈ Set.Ico a b, g' x ≤ φ x)
Goal: g b - g a ≤ ∫ (y : ℝ) in a..b, φ y
-- intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le
theorem sub_le_integral_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x) (φint : IntegrableOn φ (Icc a b))
    (hφg : ∀ x ∈ Ioo a b, g' x ≤ φ x) : g b - g a ≤ ∫ y in a..b, φ y 
Arguments: {g' g φ : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivWithinAt g (g' x) (Set.Ioi x) x) (φint : MeasureTheory.IntegrableOn φ (Set.Icc a b) MeasureTheory.volume) (hφg : ∀ x ∈ Set.Ioo a b, g' x ≤ φ x)
Goal: g b - g a ≤ ∫ (y : ℝ) in a..b, φ y
-- intervalIntegral.integral_le_sub_of_hasDeriv_right_of_le
theorem integral_le_sub_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn g (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x) (φint : IntegrableOn φ (Icc a b))
    (hφg : ∀ x ∈ Ioo a b, φ x ≤ g' x) : (∫ y in a..b, φ y) ≤ g b - g a 
Arguments: {g' g φ : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivWithinAt g (g' x) (Set.Ioi x) x) (φint : MeasureTheory.IntegrableOn φ (Set.Icc a b) MeasureTheory.volume) (hφg : ∀ x ∈ Set.Ioo a b, φ x ≤ g' x)
Goal: ∫ (y : ℝ) in a..b, φ y ≤ g b - g a
-- intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le_real
theorem integral_eq_sub_of_hasDeriv_right_of_le_real (hab : a ≤ b)
    (hcont : ContinuousOn g (Icc a b)) (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x)
    (g'int : IntegrableOn g' (Icc a b)) : ∫ y in a..b, g' y = g b - g a 
Arguments: {g' g : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b) (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivWithinAt g (g' x) (Set.Ioi x) x) (g'int : MeasureTheory.IntegrableOn g' (Set.Icc a b) MeasureTheory.volume)
Goal: ∫ (y : ℝ) in a..b, g' y = g b - g a
-- intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le
theorem integral_eq_sub_of_hasDeriv_right_of_le (hab : a ≤ b) (hcont : ContinuousOn f (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt f (f' x) (Ioi x) x)
    (f'int : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → E} (hab : a ≤ b) (hcont : ContinuousOn f (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivWithinAt f (f' x) (Set.Ioi x) x) (f'int : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (y : ℝ) in a..b, f' y = f b - f a
-- intervalIntegral.integral_eq_sub_of_hasDeriv_right
theorem integral_eq_sub_of_hasDeriv_right (hcont : ContinuousOn f (uIcc a b))
    (hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
    (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → E} (hcont : ContinuousOn f [[a, b]]) (hderiv : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x) (hint : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (y : ℝ) in a..b, f' y = f b - f a
-- intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le
theorem integral_eq_sub_of_hasDerivAt_of_le (hab : a ≤ b) (hcont : ContinuousOn f (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) :
    ∫ y in a..b, f' y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → E} (hab : a ≤ b) (hcont : ContinuousOn f (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (y : ℝ) in a..b, f' y = f b - f a
-- intervalIntegral.integral_eq_sub_of_hasDerivAt
theorem integral_eq_sub_of_hasDerivAt (hderiv : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → E} (hderiv : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (y : ℝ) in a..b, f' y = f b - f a
-- intervalIntegral.integral_eq_sub_of_hasDerivAt_of_tendsto
theorem integral_eq_sub_of_hasDerivAt_of_tendsto (hab : a < b) {fa fb}
    (hderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b)
    (ha : Tendsto f (𝓝[>] a) (𝓝 fa)) (hb : Tendsto f (𝓝[<] b) (𝓝 fb)) :
    ∫ y in a..b, f' y = fb - fa 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f f' : ℝ → E} (hab : a < b) {fa fb : E} (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' MeasureTheory.volume a b) (ha : Filter.Tendsto f (𝓝[>] a) (𝓝 fa)) (hb : Filter.Tendsto f (𝓝[<] b) (𝓝 fb))
Goal: ∫ (y : ℝ) in a..b, f' y = fb - fa
-- intervalIntegral.integral_deriv_eq_sub
theorem integral_deriv_eq_sub (hderiv : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x)
    (hint : IntervalIntegrable (deriv f) volume a b) : ∫ y in a..b, deriv f y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f : ℝ → E} (hderiv : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x) (hint : IntervalIntegrable (deriv f) MeasureTheory.volume a b)
Goal: ∫ (y : ℝ) in a..b, deriv f y = f b - f a
-- intervalIntegral.integral_deriv_eq_sub'
theorem integral_deriv_eq_sub' (f) (hderiv : deriv f = f')
    (hdiff : ∀ x ∈ uIcc a b, DifferentiableAt ℝ f x) (hcont : ContinuousOn f' (uIcc a b)) :
    ∫ y in a..b, f' y = f b - f a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} [CompleteSpace E] {f' : ℝ → E} (f : ℝ → E) (hderiv : deriv f = f') (hdiff : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x) (hcont : ContinuousOn f' [[a, b]])
Goal: ∫ (y : ℝ) in a..b, f' y = f b - f a
-- intervalIntegral.integral_unitInterval_deriv_eq_sub
lemma integral_unitInterval_deriv_eq_sub [RCLike 𝕜] [NormedSpace 𝕜 E] [IsScalarTower ℝ 𝕜 E]
    {f f' : 𝕜 → E} {z₀ z₁ : 𝕜}
    (hcont : ContinuousOn (fun t : ℝ ↦ f' (z₀ + t • z₁)) (Set.Icc 0 1))
    (hderiv : ∀ t ∈ Set.Icc (0 : ℝ) 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)) :
    z₁ • ∫ t in (0 : ℝ)..1, f' (z₀ + t • z₁) = f (z₀ + z₁) - f z₀ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [RCLike 𝕜] [NormedSpace 𝕜 E] [IsScalarTower ℝ 𝕜 E] {f f' : 𝕜 → E} {z₀ z₁ : 𝕜} (hcont : ContinuousOn (fun t => f' (z₀ + t • z₁)) (Set.Icc 0 1)) (hderiv : ∀ t ∈ Set.Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁))
Goal: z₁ • ∫ (t : ℝ) in 0 ..1, f' (z₀ + t • z₁) = f (z₀ + z₁) - f z₀
-- intervalIntegral.integrableOn_deriv_right_of_nonneg
theorem integrableOn_deriv_right_of_nonneg (hcont : ContinuousOn g (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x)
    (g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) : IntegrableOn g' (Ioc a b) 
Arguments: {g' g : ℝ → ℝ} {a b : ℝ} (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivWithinAt g (g' x) (Set.Ioi x) x) (g'pos : ∀ x ∈ Set.Ioo a b, 0 ≤ g' x)
Goal: MeasureTheory.IntegrableOn g' (Set.Ioc a b) MeasureTheory.volume
-- intervalIntegral.integrableOn_deriv_of_nonneg
theorem integrableOn_deriv_of_nonneg (hcont : ContinuousOn g (Icc a b))
    (hderiv : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) :
    IntegrableOn g' (Ioc a b) 
Arguments: {g' g : ℝ → ℝ} {a b : ℝ} (hcont : ContinuousOn g (Set.Icc a b)) (hderiv : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Set.Ioo a b, 0 ≤ g' x)
Goal: MeasureTheory.IntegrableOn g' (Set.Ioc a b) MeasureTheory.volume
-- intervalIntegral.intervalIntegrable_deriv_of_nonneg
theorem intervalIntegrable_deriv_of_nonneg (hcont : ContinuousOn g (uIcc a b))
    (hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x)
    (hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) : IntervalIntegrable g' volume a b 
Arguments: {g' g : ℝ → ℝ} {a b : ℝ} (hcont : ContinuousOn g [[a, b]]) (hderiv : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivAt g (g' x) x) (hpos : ∀ x ∈ Set.Ioo (min a b) (max a b), 0 ≤ g' x)
Goal: IntervalIntegrable g' MeasureTheory.volume a b