ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AECover.integrable_of_integral_norm_bounded
theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (φ i) μ) (hbounded : ∀ᶠ (i : ι) in l, ∫ (x : α) in φ i, ‖f x‖ ∂μ ≤ I)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_integral_norm_tendsto
theorem AECover.integrable_of_integral_norm_tendsto [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (htendsto : Tendsto (fun i => ∫ x in φ i, ‖f x‖ ∂μ) l (𝓝 I)) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (φ i) μ) (htendsto : Filter.Tendsto (fun i => ∫ (x : α) in φ i, ‖f x‖ ∂μ) l (𝓝 I))
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
theorem AECover.integrable_of_integral_bounded_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ i in l, (∫ x in φ i, f x ∂μ) ≤ I) : Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (φ i) μ) (hnng : ∀ᵐ (x : α) ∂μ, 0 ≤ f x) (hbounded : ∀ᶠ (i : ι) in l, ∫ (x : α) in φ i, f x ∂μ ≤ I)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integrable_of_integral_tendsto_of_nonneg_ae
theorem AECover.integrable_of_integral_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
    (hnng : ∀ᵐ x ∂μ, 0 ≤ f x) (htendsto : Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 I)) :
    Integrable f μ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (φ i) μ) (hnng : ∀ᵐ (x : α) ∂μ, 0 ≤ f x) (htendsto : Filter.Tendsto (fun i => ∫ (x : α) in φ i, f x ∂μ) l (𝓝 I))
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.AECover.integral_tendsto_of_countably_generated
theorem AECover.integral_tendsto_of_countably_generated [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → E} (hfi : Integrable f μ) :
    Tendsto (fun i => ∫ x in φ i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [NormedSpace ℝ E] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (hfi : MeasureTheory.Integrable f μ)
Goal: Filter.Tendsto (fun i => ∫ (x : α) in φ i, f x ∂μ) l (𝓝 (∫ (x : α), f x ∂μ))
-- MeasureTheory.AECover.integral_eq_of_tendsto
theorem AECover.integral_eq_of_tendsto [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α}
    (hφ : AECover μ l φ) {f : α → E} (I : E) (hfi : Integrable f μ)
    (h : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) : ∫ x, f x ∂μ = I 
Arguments: {α : Type u_1} {ι : Type u_2} {E : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [NormedAddCommGroup E] [NormedSpace ℝ E] [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → E} (I : E) (hfi : MeasureTheory.Integrable f μ) (h : Filter.Tendsto (fun n => ∫ (x : α) in φ n, f x ∂μ) l (𝓝 I))
Goal: ∫ (x : α), f x ∂μ = I
-- MeasureTheory.AECover.integral_eq_of_tendsto_of_nonneg_ae
theorem AECover.integral_eq_of_tendsto_of_nonneg_ae [l.NeBot] [l.IsCountablyGenerated]
    {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f)
    (hfi : ∀ n, IntegrableOn f (φ n) μ) (htendsto : Tendsto (fun n => ∫ x in φ n, f x ∂μ) l (𝓝 I)) :
    ∫ x, f x ∂μ = I 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) {f : α → ℝ} (I : ℝ) (hnng : 0 ≤ᵐ[μ] f) (hfi : ∀ (n : ι), MeasureTheory.IntegrableOn f (φ n) μ) (htendsto : Filter.Tendsto (fun n => ∫ (x : α) in φ n, f x ∂μ) l (𝓝 I))
Goal: ∫ (x : α), f x ∂μ = I
-- MeasureTheory.integrable_of_intervalIntegral_norm_bounded
theorem integrable_of_intervalIntegral_norm_bounded (I : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) (h : ∀ᶠ i in l, (∫ x in a i..b i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) (b i)) μ) (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ ≤ I)
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integrable_of_intervalIntegral_norm_tendsto
theorem integrable_of_intervalIntegral_norm_tendsto (I : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) (b i)) μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) (h : Tendsto (fun i => ∫ x in a i..b i, ‖f x‖ ∂μ) l (𝓝 I)) :
    Integrable f μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E} (I : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) (b i)) μ) (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop) (h : Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b i, ‖f x‖ ∂μ) l (𝓝 I))
Goal: MeasureTheory.Integrable f μ
-- MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_bounded
theorem integrableOn_Iic_of_intervalIntegral_norm_bounded (I b : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
    (h : ∀ᶠ i in l, (∫ x in a i..b, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Iic b) μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a : ι → ℝ} {f : ℝ → E} (I b : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) b) μ) (ha : Filter.Tendsto a l Filter.atBot) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I)
Goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ
-- MeasureTheory.integrableOn_Iic_of_intervalIntegral_norm_tendsto
theorem integrableOn_Iic_of_intervalIntegral_norm_tendsto (I b : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc (a i) b) μ) (ha : Tendsto a l atBot)
    (h : Tendsto (fun i => ∫ x in a i..b, ‖f x‖ ∂μ) l (𝓝 I)) : IntegrableOn f (Iic b) μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a : ι → ℝ} {f : ℝ → E} (I b : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) b) μ) (ha : Filter.Tendsto a l Filter.atBot) (h : Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ) l (𝓝 I))
Goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ
-- MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_bounded
theorem integrableOn_Ioi_of_intervalIntegral_norm_bounded (I a : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
    (h : ∀ᶠ i in l, (∫ x in a..b i, ‖f x‖ ∂μ) ≤ I) : IntegrableOn f (Ioi a) μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {b : ι → ℝ} {f : ℝ → E} (I a : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc a (b i)) μ) (hb : Filter.Tendsto b l Filter.atTop) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ ≤ I)
Goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ
-- MeasureTheory.integrableOn_Ioi_of_intervalIntegral_norm_tendsto
theorem integrableOn_Ioi_of_intervalIntegral_norm_tendsto (I a : ℝ)
    (hfi : ∀ i, IntegrableOn f (Ioc a (b i)) μ) (hb : Tendsto b l atTop)
    (h : Tendsto (fun i => ∫ x in a..b i, ‖f x‖ ∂μ) l (𝓝 <| I)) : IntegrableOn f (Ioi a) μ 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {b : ι → ℝ} {f : ℝ → E} (I a : ℝ) (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc a (b i)) μ) (hb : Filter.Tendsto b l Filter.atTop) (h : Filter.Tendsto (fun i => ∫ (x : ℝ) in a..b i, ‖f x‖ ∂μ) l (𝓝 I))
Goal: MeasureTheory.IntegrableOn f (Set.Ioi a) μ
-- MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded
theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded {I a₀ b₀ : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc (a i) (b i)) (ha : Tendsto a l <| 𝓝 a₀)
    (hb : Tendsto b l <| 𝓝 b₀) (h : ∀ᶠ i in l, (∫ x in Ioc (a i) (b i), ‖f x‖) ≤ I) :
    IntegrableOn f (Ioc a₀ b₀) 
Arguments: {ι : Type u_1} {E : Type u_2} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a b : ι → ℝ} {f : ℝ → E} {I a₀ b₀ : ℝ} (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) (b i)) MeasureTheory.volume) (ha : Filter.Tendsto a l (𝓝 a₀)) (hb : Filter.Tendsto b l (𝓝 b₀)) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Set.Ioc (a i) (b i), ‖f x‖ ≤ I)
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a₀ b₀) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_left
theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_left {I a₀ b : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc (a i) b) (ha : Tendsto a l <| 𝓝 a₀)
    (h : ∀ᶠ i in l, (∫ x in Ioc (a i) b, ‖f x‖) ≤ I) : IntegrableOn f (Ioc a₀ b) 
Arguments: {ι : Type u_1} {E : Type u_2} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {a : ι → ℝ} {f : ℝ → E} {I a₀ b : ℝ} (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc (a i) b) MeasureTheory.volume) (ha : Filter.Tendsto a l (𝓝 a₀)) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Set.Ioc (a i) b, ‖f x‖ ≤ I)
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a₀ b) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioc_of_intervalIntegral_norm_bounded_right
theorem integrableOn_Ioc_of_intervalIntegral_norm_bounded_right {I a b₀ : ℝ}
    (hfi : ∀ i, IntegrableOn f <| Ioc a (b i)) (hb : Tendsto b l <| 𝓝 b₀)
    (h : ∀ᶠ i in l, (∫ x in Ioc a (b i), ‖f x‖) ≤ I) : IntegrableOn f (Ioc a b₀) 
Arguments: {ι : Type u_1} {E : Type u_2} {l : Filter ι} [l.NeBot] [l.IsCountablyGenerated] [NormedAddCommGroup E] {b : ι → ℝ} {f : ℝ → E} {I a b₀ : ℝ} (hfi : ∀ (i : ι), MeasureTheory.IntegrableOn f (Set.Ioc a (b i)) MeasureTheory.volume) (hb : Filter.Tendsto b l (𝓝 b₀)) (h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Set.Ioc a (b i), ‖f x‖ ≤ I)
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a b₀) MeasureTheory.volume
-- MeasureTheory.intervalIntegral_tendsto_integral
theorem intervalIntegral_tendsto_integral (hfi : Integrable f μ) (ha : Tendsto a l atBot)
    (hb : Tendsto b l atTop) : Tendsto (fun i => ∫ x in a i..b i, f x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.IsCountablyGenerated] [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ι → ℝ} {f : ℝ → E} (hfi : MeasureTheory.Integrable f μ) (ha : Filter.Tendsto a l Filter.atBot) (hb : Filter.Tendsto b l Filter.atTop)
Goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ), f x ∂μ))
-- MeasureTheory.intervalIntegral_tendsto_integral_Iic
theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
    (ha : Tendsto a l atBot) :
    Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.IsCountablyGenerated] [NormedAddCommGroup E] [NormedSpace ℝ E] {a : ι → ℝ} {f : ℝ → E} (b : ℝ) (hfi : MeasureTheory.IntegrableOn f (Set.Iic b) μ) (ha : Filter.Tendsto a l Filter.atBot)
Goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Set.Iic b, f x ∂μ))
-- MeasureTheory.intervalIntegral_tendsto_integral_Ioi
theorem intervalIntegral_tendsto_integral_Ioi (a : ℝ) (hfi : IntegrableOn f (Ioi a) μ)
    (hb : Tendsto b l atTop) :
    Tendsto (fun i => ∫ x in a..b i, f x ∂μ) l (𝓝 <| ∫ x in Ioi a, f x ∂μ) 
Arguments: {ι : Type u_1} {E : Type u_2} {μ : MeasureTheory.Measure ℝ} {l : Filter ι} [l.IsCountablyGenerated] [NormedAddCommGroup E] [NormedSpace ℝ E] {b : ι → ℝ} {f : ℝ → E} (a : ℝ) (hfi : MeasureTheory.IntegrableOn f (Set.Ioi a) μ) (hb : Filter.Tendsto b l Filter.atTop)
Goal: Filter.Tendsto (fun i => ∫ (x : ℝ) in a..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Set.Ioi a, f x ∂μ))
-- MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi
theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Ioi [CompleteSpace E]
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a)) :
    Tendsto f atTop (𝓝 (limUnder atTop f)) 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume)
Goal: Filter.Tendsto f Filter.atTop (𝓝 (limUnder Filter.atTop f))
-- MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Ioi a)) (fint : IntegrableOn f (Ioi a)) :
    Tendsto f atTop (𝓝 0) 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} [NormedAddCommGroup E] [NormedSpace ℝ E] (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume) (fint : MeasureTheory.IntegrableOn f (Set.Ioi a) MeasureTheory.volume)
Goal: Filter.Tendsto f Filter.atTop (𝓝 0)
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto
theorem integral_Ioi_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Ioi a))
    (hf : Tendsto f atTop (𝓝 m)) : ∫ x in Ioi a, f' x = m - f a 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hcont : ContinuousWithinAt f (Set.Ici a) a) (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume) (hf : Filter.Tendsto f Filter.atTop (𝓝 m))
Goal: ∫ (x : ℝ) in Set.Ioi a, f' x = m - f a
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_tendsto'
theorem integral_Ioi_of_hasDerivAt_of_tendsto' (hderiv : ∀ x ∈ Ici a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Ioi a)) (hf : Tendsto f atTop (𝓝 m)) :
    ∫ x in Ioi a, f' x = m - f a 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hderiv : ∀ x ∈ Set.Ici a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume) (hf : Filter.Tendsto f Filter.atTop (𝓝 m))
Goal: ∫ (x : ℝ) in Set.Ioi a, f' x = m - f a
-- HasCompactSupport.integral_Ioi_deriv_eq
theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b 
Arguments: {E : Type u_1} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f) (b : ℝ)
Goal: ∫ (x : ℝ) in Set.Ioi b, deriv f x = -f b
-- MeasureTheory.integrableOn_Ioi_deriv_of_nonneg
theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hcont : ContinuousWithinAt g (Set.Ici a) a) (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Set.Ioi a, 0 ≤ g' x) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioi_deriv_of_nonneg'
theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hderiv : ∀ x ∈ Set.Ici a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Set.Ioi a, 0 ≤ g' x) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg
theorem integral_Ioi_of_hasDerivAt_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hcont : ContinuousWithinAt g (Set.Ici a) a) (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Set.Ioi a, 0 ≤ g' x) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonneg'
theorem integral_Ioi_of_hasDerivAt_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hderiv : ∀ x ∈ Set.Ici a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Set.Ioi a, 0 ≤ g' x) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a
-- MeasureTheory.integrableOn_Ioi_deriv_of_nonpos
theorem integrableOn_Ioi_deriv_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
    (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hcont : ContinuousWithinAt g (Set.Ici a) a) (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Set.Ioi a, g' x ≤ 0) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume
-- MeasureTheory.integrableOn_Ioi_deriv_of_nonpos'
theorem integrableOn_Ioi_deriv_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hderiv : ∀ x ∈ Set.Ici a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Set.Ioi a, g' x ≤ 0) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: MeasureTheory.IntegrableOn g' (Set.Ioi a) MeasureTheory.volume
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos
theorem integral_Ioi_of_hasDerivAt_of_nonpos (hcont : ContinuousWithinAt g (Ici a) a)
    (hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0)
    (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hcont : ContinuousWithinAt g (Set.Ici a) a) (hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Set.Ioi a, g' x ≤ 0) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a
-- MeasureTheory.integral_Ioi_of_hasDerivAt_of_nonpos'
theorem integral_Ioi_of_hasDerivAt_of_nonpos' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
    (g'neg : ∀ x ∈ Ioi a, g' x ≤ 0) (hg : Tendsto g atTop (𝓝 l)) : ∫ x in Ioi a, g' x = l - g a 
Arguments: {g g' : ℝ → ℝ} {a l : ℝ} (hderiv : ∀ x ∈ Set.Ici a, HasDerivAt g (g' x) x) (g'neg : ∀ x ∈ Set.Ioi a, g' x ≤ 0) (hg : Filter.Tendsto g Filter.atTop (𝓝 l))
Goal: ∫ (x : ℝ) in Set.Ioi a, g' x = l - g a
-- MeasureTheory.tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic
theorem tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic [CompleteSpace E]
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a)) :
    Tendsto f atBot (𝓝 (limUnder atBot f)) 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hderiv : ∀ x ∈ Set.Iic a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Iic a) MeasureTheory.volume)
Goal: Filter.Tendsto f Filter.atBot (𝓝 (limUnder Filter.atBot f))
-- MeasureTheory.tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x)
    (f'int : IntegrableOn f' (Iic a)) (fint : IntegrableOn f (Iic a)) :
    Tendsto f atBot (𝓝 0) 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} [NormedAddCommGroup E] [NormedSpace ℝ E] (hderiv : ∀ x ∈ Set.Iic a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Iic a) MeasureTheory.volume) (fint : MeasureTheory.IntegrableOn f (Set.Iic a) MeasureTheory.volume)
Goal: Filter.Tendsto f Filter.atBot (𝓝 0)
-- MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto
theorem integral_Iic_of_hasDerivAt_of_tendsto (hcont : ContinuousWithinAt f (Iic a) a)
    (hderiv : ∀ x ∈ Iio a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hcont : ContinuousWithinAt f (Set.Iic a) a) (hderiv : ∀ x ∈ Set.Iio a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Iic a) MeasureTheory.volume) (hf : Filter.Tendsto f Filter.atBot (𝓝 m))
Goal: ∫ (x : ℝ) in Set.Iic a, f' x = f a - m
-- MeasureTheory.integral_Iic_of_hasDerivAt_of_tendsto'
theorem integral_Iic_of_hasDerivAt_of_tendsto'
    (hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
    (hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m 
Arguments: {E : Type u_1} {f f' : ℝ → E} {a : ℝ} {m : E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hderiv : ∀ x ∈ Set.Iic a, HasDerivAt f (f' x) x) (f'int : MeasureTheory.IntegrableOn f' (Set.Iic a) MeasureTheory.volume) (hf : Filter.Tendsto f Filter.atBot (𝓝 m))
Goal: ∫ (x : ℝ) in Set.Iic a, f' x = f a - m
-- HasCompactSupport.integral_Iic_deriv_eq
theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
    (h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Iic b, deriv f x = f b 
Arguments: {E : Type u_1} {f : ℝ → E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hf : ContDiff ℝ 1 f) (h2f : HasCompactSupport f) (b : ℝ)
Goal: ∫ (x : ℝ) in Set.Iic b, deriv f x = f b
-- HasCompactSupport.ennnorm_le_lintegral_Ici_deriv
lemma _root_.HasCompactSupport.ennnorm_le_lintegral_Ici_deriv
    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
    {f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
    (‖f x‖₊ : ℝ≥0∞) ≤ ∫⁻ y in Iic x, ‖deriv f y‖₊ 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ)
Goal: ↑‖f x‖₊ ≤ ∫⁻ (y : ℝ) in Set.Iic x, ↑‖deriv f y‖₊
-- MeasureTheory.integral_of_hasDerivAt_of_tendsto
theorem integral_of_hasDerivAt_of_tendsto [CompleteSpace E]
    (hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f')
    (hbot : Tendsto f atBot (𝓝 m)) (htop : Tendsto f atTop (𝓝 n)) : ∫ x, f' x = n - m 
Arguments: {E : Type u_1} {f f' : ℝ → E} {m n : E} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (hderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : MeasureTheory.Integrable f' MeasureTheory.volume) (hbot : Filter.Tendsto f Filter.atBot (𝓝 m)) (htop : Filter.Tendsto f Filter.atTop (𝓝 n))
Goal: ∫ (x : ℝ), f' x = n - m