ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.integral_eq_of_hasDerivWithinAt_off_countable
theorem integral_eq_of_hasDerivWithinAt_off_countable (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ}
    (hs : s.Countable) (Hc : ContinuousOn f [[a, b]])
    (Hd : ∀ x ∈ Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x)
    (Hi : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f f' : ℝ → E) {a b : ℝ} {s : Set ℝ} (hs : s.Countable) (Hc : ContinuousOn f [[a, b]]) (Hd : ∀ x ∈ Set.Ioo (min a b) (max a b) \ s, HasDerivAt f (f' x) x) (Hi : IntervalIntegrable f' MeasureTheory.volume a b)
Goal: ∫ (x : ℝ) in a..b, f' x = f b - f a
-- MeasureTheory.integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le
theorem integral_divergence_prod_Icc_of_hasFDerivWithinAt_off_countable_of_le (f g : ℝ × ℝ → E)
    (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a b : ℝ × ℝ) (hle : a ≤ b) (s : Set (ℝ × ℝ)) (hs : s.Countable)
    (Hcf : ContinuousOn f (Icc a b)) (Hcg : ContinuousOn g (Icc a b))
    (Hdf : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x)
    (Hdg : ∀ x ∈ Ioo a.1 b.1 ×ˢ Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x)
    (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) (Icc a b)) :
    (∫ x in Icc a b, f' x (1, 0) + g' x (0, 1)) =
      (((∫ x in a.1..b.1, g (x, b.2)) - ∫ x in a.1..b.1, g (x, a.2)) +
          ∫ y in a.2..b.2, f (b.1, y)) -
        ∫ y in a.2..b.2, f (a.1, y) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f g : ℝ × ℝ → E) (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a b : ℝ × ℝ) (hle : a ≤ b) (s : Set (ℝ × ℝ)) (hs : s.Countable) (Hcf : ContinuousOn f (Set.Icc a b)) (Hcg : ContinuousOn g (Set.Icc a b)) (Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt f (f' x) x) (Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \ s, HasFDerivAt g (g' x) x) (Hi : MeasureTheory.IntegrableOn (fun x => (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) MeasureTheory.volume)
Goal: ∫ (x : ℝ × ℝ) in Set.Icc a b, (f' x) (1, 0) + (g' x) (0, 1) =
  (((∫ (x : ℝ) in a.1 ..b.1, g (x, b.2)) - ∫ (x : ℝ) in a.1 ..b.1, g (x, a.2)) + ∫ (y : ℝ) in a.2 ..b.2, f (b.1, y)) -
    ∫ (y : ℝ) in a.2 ..b.2, f (a.1, y)
-- MeasureTheory.integral2_divergence_prod_of_hasFDerivWithinAt_off_countable
theorem integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (f g : ℝ × ℝ → E)
    (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a₁ a₂ b₁ b₂ : ℝ) (s : Set (ℝ × ℝ)) (hs : s.Countable)
    (Hcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]]))
    (Hcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]]))
    (Hdf : ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s,
      HasFDerivAt f (f' x) x)
    (Hdg : ∀ x ∈ Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Ioo (min a₂ b₂) (max a₂ b₂) \ s,
      HasFDerivAt g (g' x) x)
    (Hi : IntegrableOn (fun x => f' x (1, 0) + g' x (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]])) :
    (∫ x in a₁..b₁, ∫ y in a₂..b₂, f' (x, y) (1, 0) + g' (x, y) (0, 1)) =
      (((∫ x in a₁..b₁, g (x, b₂)) - ∫ x in a₁..b₁, g (x, a₂)) + ∫ y in a₂..b₂, f (b₁, y)) -
        ∫ y in a₂..b₂, f (a₁, y) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (f g : ℝ × ℝ → E) (f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E) (a₁ a₂ b₁ b₂ : ℝ) (s : Set (ℝ × ℝ)) (hs : s.Countable) (Hcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])) (Hcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])) (Hdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt f (f' x) x) (Hdg : ∀ x ∈ Set.Ioo (min a₁ b₁) (max a₁ b₁) ×ˢ Set.Ioo (min a₂ b₂) (max a₂ b₂) \ s, HasFDerivAt g (g' x) x) (Hi :
  MeasureTheory.IntegrableOn (fun x => (f' x) (1, 0) + (g' x) (0, 1)) ([[a₁, b₁]] ×ˢ [[a₂, b₂]]) MeasureTheory.volume)
Goal: ∫ (x : ℝ) in a₁..b₁, ∫ (y : ℝ) in a₂..b₂, (f' (x, y)) (1, 0) + (g' (x, y)) (0, 1) =
  (((∫ (x : ℝ) in a₁..b₁, g (x, b₂)) - ∫ (x : ℝ) in a₁..b₁, g (x, a₂)) + ∫ (y : ℝ) in a₂..b₂, f (b₁, y)) -
    ∫ (y : ℝ) in a₂..b₂, f (a₁, y)
-- Real.measurable_exp
@[measurability]
theorem measurable_exp : Measurable exp 
Goal: Measurable Real.exp
-- Real.measurable_log
@[measurability]
theorem measurable_log : Measurable log 
Goal: Measurable Real.log
-- Real.measurable_of_measurable_exp
lemma measurable_of_measurable_exp {α : Type*} {_ : MeasurableSpace α} {f : α → ℝ}
    (hf : Measurable (fun x ↦ exp (f x))) :
    Measurable f 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {f : α → ℝ}, (Measurable fun x => Real.exp (f x)) → Measurable f
-- Real.aemeasurable_of_aemeasurable_exp
lemma aemeasurable_of_aemeasurable_exp {α : Type*} {_ : MeasurableSpace α} {f : α → ℝ}
    {μ : MeasureTheory.Measure α} (hf : AEMeasurable (fun x ↦ exp (f x)) μ) :
    AEMeasurable f μ 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {f : α → ℝ} {μ : MeasureTheory.Measure α},
  AEMeasurable (fun x => Real.exp (f x)) μ → AEMeasurable f μ
-- Real.measurable_sin
@[measurability]
theorem measurable_sin : Measurable sin 
Goal: Measurable Real.sin
-- Real.measurable_cos
@[measurability]
theorem measurable_cos : Measurable cos 
Goal: Measurable Real.cos
-- Real.measurable_sinh
@[measurability]
theorem measurable_sinh : Measurable sinh 
Goal: Measurable Real.sinh
-- Real.measurable_cosh
@[measurability]
theorem measurable_cosh : Measurable cosh 
Goal: Measurable Real.cosh
-- Real.measurable_arcsin
@[measurability]
theorem measurable_arcsin : Measurable arcsin 
Goal: Measurable Real.arcsin
-- Real.measurable_arccos
@[measurability]
theorem measurable_arccos : Measurable arccos 
Goal: Measurable Real.arccos
-- Complex.measurable_re
@[measurability]
theorem measurable_re : Measurable re 
Goal: Measurable Complex.re
-- Complex.measurable_im
@[measurability]
theorem measurable_im : Measurable im 
Goal: Measurable Complex.im
-- Complex.measurable_ofReal
@[measurability]
theorem measurable_ofReal : Measurable ((↑) : ℝ → ℂ) 
Goal: Measurable Complex.ofReal'
-- Complex.measurable_exp
@[measurability]
theorem measurable_exp : Measurable exp 
Goal: Measurable Complex.exp
-- Complex.measurable_sin
@[measurability]
theorem measurable_sin : Measurable sin 
Goal: Measurable Complex.sin
-- Complex.measurable_cos
@[measurability]
theorem measurable_cos : Measurable cos 
Goal: Measurable Complex.cos
-- Complex.measurable_sinh
@[measurability]
theorem measurable_sinh : Measurable sinh 
Goal: Measurable Complex.sinh
-- Complex.measurable_cosh
@[measurability]
theorem measurable_cosh : Measurable cosh 
Goal: Measurable Complex.cosh
-- Complex.measurable_arg
@[measurability]
theorem measurable_arg : Measurable arg 
Goal: Measurable Complex.arg
-- Complex.measurable_log
@[measurability]
theorem measurable_log : Measurable log 
Goal: Measurable Complex.log
-- Measurable.exp
@[measurability]
theorem Measurable.exp : Measurable fun x => Real.exp (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.exp (f x)
-- Measurable.log
@[measurability]
theorem Measurable.log : Measurable fun x => log (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.log (f x)
-- Measurable.cos
@[measurability]
theorem Measurable.cos : Measurable fun x => Real.cos (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.cos (f x)
-- Measurable.sin
@[measurability]
theorem Measurable.sin : Measurable fun x => Real.sin (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.sin (f x)
-- Measurable.cosh
@[measurability]
theorem Measurable.cosh : Measurable fun x => Real.cosh (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.cosh (f x)
-- Measurable.sinh
@[measurability]
theorem Measurable.sinh : Measurable fun x => Real.sinh (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => Real.sinh (f x)
-- Measurable.sqrt
@[measurability]
theorem Measurable.sqrt : Measurable fun x => √(f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => √(f x)
-- Measurable.cexp
@[measurability]
theorem Measurable.cexp : Measurable fun x => Complex.exp (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.exp (f x)
-- Measurable.ccos
@[measurability]
theorem Measurable.ccos : Measurable fun x => Complex.cos (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.cos (f x)
-- Measurable.csin
@[measurability]
theorem Measurable.csin : Measurable fun x => Complex.sin (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.sin (f x)
-- Measurable.ccosh
@[measurability]
theorem Measurable.ccosh : Measurable fun x => Complex.cosh (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.cosh (f x)
-- Measurable.csinh
@[measurability]
theorem Measurable.csinh : Measurable fun x => Complex.sinh (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.sinh (f x)
-- Measurable.carg
@[measurability]
theorem Measurable.carg : Measurable fun x => arg (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => (f x).arg
-- Measurable.clog
@[measurability]
theorem Measurable.clog : Measurable fun x => Complex.log (f x) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℂ} (hf : Measurable f)
Goal: Measurable fun x => Complex.log (f x)
-- integrableOn_peak_smul_of_integrableOn_of_tendsto
theorem integrableOn_peak_smul_of_integrableOn_of_tendsto
    (hs : MeasurableSet s) (h'st : t ∈ 𝓝[s] x₀)
    (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
    (hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
    (h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
    (hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)) :
    ∀ᶠ i in l, IntegrableOn (fun x => φ i x • g x) s μ 
Arguments: {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s t : Set α} {φ : ι → α → ℝ} {a : E} (hs : MeasurableSet s) (h'st : t ∈ 𝓝[s] x₀) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : Filter.Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)) (h'iφ : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (φ i) (μ.restrict s)) (hmg : MeasureTheory.IntegrableOn g s μ) (hcg : Filter.Tendsto g (𝓝[s] x₀) (𝓝 a))
Goal: ∀ᶠ (i : ι) in l, MeasureTheory.IntegrableOn (fun x => φ i x • g x) s μ
-- tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux
theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux
    (hs : MeasurableSet s) (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
    (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
    (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
    (hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
    (h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
    (hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 0)) :
    Tendsto (fun i : ι => ∫ x in s, φ i x • g x ∂μ) l (𝓝 0) 
Arguments: {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s t : Set α} {φ : ι → α → ℝ} (hs : MeasurableSet s) (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀) (hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : Filter.Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)) (h'iφ : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (φ i) (μ.restrict s)) (hmg : MeasureTheory.IntegrableOn g s μ) (hcg : Filter.Tendsto g (𝓝[s] x₀) (𝓝 0))
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s, φ i x • g x ∂μ) l (𝓝 0)
-- tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
    (hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
    (h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
    (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
    (hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
    (h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
    (hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)) :
    Tendsto (fun i : ι ↦ ∫ x in s, φ i x • g x ∂μ) l (𝓝 a) 
Arguments: {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {s : Set α} {φ : ι → α → ℝ} {a : E} [CompleteSpace E] (hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀) (h't : μ t ≠ ⊤) (hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)) (hiφ : Filter.Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)) (h'iφ : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (φ i) (μ.restrict s)) (hmg : MeasureTheory.IntegrableOn g s μ) (hcg : Filter.Tendsto g (𝓝[s] x₀) (𝓝 a))
Goal: Filter.Tendsto (fun i => ∫ (x : α) in s, φ i x • g x ∂μ) l (𝓝 a)
-- tendsto_integral_peak_smul_of_integrable_of_tendsto
theorem tendsto_integral_peak_smul_of_integrable_of_tendsto
    {t : Set α} (ht : MeasurableSet t) (h'ts : t ∈ 𝓝 x₀)
    (h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
    (hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l uᶜ)
    (hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
    (h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) μ)
    (hmg : Integrable g μ) (hcg : Tendsto g (𝓝 x₀) (𝓝 a)) :
    Tendsto (fun i : ι ↦ ∫ x, φ i x • g x ∂μ) l (𝓝 a) 
Arguments: {α : Type u_1} {E : Type u_2} {ι : Type u_3} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {l : Filter ι} {x₀ : α} {φ : ι → α → ℝ} {a : E} [CompleteSpace E] {t : Set α} (ht : MeasurableSet t) (h'ts : t ∈ 𝓝 x₀) (h't : μ t ≠ ⊤) (hnφ : ∀ᶠ (i : ι) in l, ∀ (x : α), 0 ≤ φ i x) (hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l uᶜ) (hiφ : Filter.Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)) (h'iφ : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (φ i) μ) (hmg : MeasureTheory.Integrable g μ) (hcg : Filter.Tendsto g (𝓝 x₀) (𝓝 a))
Goal: Filter.Tendsto (fun i => ∫ (x : α), φ i x • g x ∂μ) l (𝓝 a)
-- tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos
theorem tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_measure_nhdsWithin_pos
    [MetrizableSpace α] [IsLocallyFiniteMeasure μ] (hs : IsCompact s)
    (hμ : ∀ u, IsOpen u → x₀ ∈ u → 0 < μ (u ∩ s)) {c : α → ℝ} (hc : ContinuousOn c s)
    (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ s)
    (hmg : IntegrableOn g s μ) (hcg : ContinuousWithinAt g s x₀) :
    Tendsto (fun n : ℕ => (∫ x in s, c x ^ n ∂μ)⁻¹ • ∫ x in s, c x ^ n • g x ∂μ)
      atTop (𝓝 (g x₀)) 
Arguments: {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] (hs : IsCompact s) (hμ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → 0 < μ (u ∩ s)) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ s) (hmg : MeasureTheory.IntegrableOn g s μ) (hcg : ContinuousWithinAt g s x₀)
Goal: Filter.Tendsto (fun n => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (𝓝 (g x₀))
-- tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn
theorem tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_integrableOn
    [MetrizableSpace α] [IsLocallyFiniteMeasure μ] [IsOpenPosMeasure μ] (hs : IsCompact s)
    {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀)
    (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s))
    (hmg : IntegrableOn g s μ) (hcg : ContinuousWithinAt g s x₀) :
    Tendsto (fun n : ℕ => (∫ x in s, c x ^ n ∂μ)⁻¹ • ∫ x in s, c x ^ n • g x ∂μ) atTop
      (𝓝 (g x₀)) 
Arguments: {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] (hs : IsCompact s) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s)) (hmg : MeasureTheory.IntegrableOn g s μ) (hcg : ContinuousWithinAt g s x₀)
Goal: Filter.Tendsto (fun n => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (𝓝 (g x₀))
-- tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn
theorem tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn
    [MetrizableSpace α] [IsLocallyFiniteMeasure μ] [IsOpenPosMeasure μ] (hs : IsCompact s)
    {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀)
    (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s))
    (hmg : ContinuousOn g s) :
    Tendsto (fun n : ℕ => (∫ x in s, c x ^ n ∂μ)⁻¹ • ∫ x in s, c x ^ n • g x ∂μ) atTop (𝓝 (g x₀)) 
Arguments: {α : Type u_1} {E : Type u_2} {hm : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [BorelSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] {g : α → E} {x₀ : α} {s : Set α} [CompleteSpace E] [TopologicalSpace.MetrizableSpace α] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] (hs : IsCompact s) {c : α → ℝ} (hc : ContinuousOn c s) (h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀) (hnc : ∀ x ∈ s, 0 ≤ c x) (hnc₀ : 0 < c x₀) (h₀ : x₀ ∈ closure (interior s)) (hmg : ContinuousOn g s)
Goal: Filter.Tendsto (fun n => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (𝓝 (g x₀))
-- tendsto_integral_comp_smul_smul_of_integrable
theorem tendsto_integral_comp_smul_smul_of_integrable
    {φ : F → ℝ} (hφ : ∀ x, 0 ≤ φ x) (h'φ : ∫ x, φ x ∂μ = 1)
    (h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0))
    {g : F → E} (hg : Integrable g μ) (h'g : ContinuousAt g 0) :
    Tendsto (fun (c : ℝ) ↦ ∫ x, (c ^ (finrank ℝ F) * φ (c • x)) • g x ∂μ) atTop (𝓝 (g 0)) 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {μ : MeasureTheory.Measure F} [μ.IsAddHaarMeasure] {φ : F → ℝ} (hφ : ∀ (x : F), 0 ≤ φ x) (h'φ : ∫ (x : F), φ x ∂μ = 1) (h : Filter.Tendsto (fun x => ‖x‖ ^ FiniteDimensional.finrank ℝ F * φ x) (Bornology.cobounded F) (𝓝 0)) {g : F → E} (hg : MeasureTheory.Integrable g μ) (h'g : ContinuousAt g 0)
Goal: Filter.Tendsto (fun c => ∫ (x : F), (c ^ FiniteDimensional.finrank ℝ F * φ (c • x)) • g x ∂μ) Filter.atTop (𝓝 (g 0))
-- tendsto_integral_comp_smul_smul_of_integrable'
theorem tendsto_integral_comp_smul_smul_of_integrable'
    {φ : F → ℝ} (hφ : ∀ x, 0 ≤ φ x) (h'φ : ∫ x, φ x ∂μ = 1)
    (h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0))
    {g : F → E} {x₀ : F} (hg : Integrable g μ) (h'g : ContinuousAt g x₀) :
    Tendsto (fun (c : ℝ) ↦ ∫ x, (c ^ (finrank ℝ F) * φ (c • (x₀ - x))) • g x ∂μ)
      atTop (𝓝 (g x₀)) 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {μ : MeasureTheory.Measure F} [μ.IsAddHaarMeasure] {φ : F → ℝ} (hφ : ∀ (x : F), 0 ≤ φ x) (h'φ : ∫ (x : F), φ x ∂μ = 1) (h : Filter.Tendsto (fun x => ‖x‖ ^ FiniteDimensional.finrank ℝ F * φ x) (Bornology.cobounded F) (𝓝 0)) {g : F → E} {x₀ : F} (hg : MeasureTheory.Integrable g μ) (h'g : ContinuousAt g x₀)
Goal: Filter.Tendsto (fun c => ∫ (x : F), (c ^ FiniteDimensional.finrank ℝ F * φ (c • (x₀ - x))) • g x ∂μ) Filter.atTop
  (𝓝 (g x₀))
-- MeasureTheory.AECover.measurableSet
structure AECover (φ : ι → Set α) : Prop where
  ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
  protected measurableSet : ∀ i, MeasurableSet <| φ i
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ : ι → Set α} (self : MeasureTheory.AECover μ l φ) (i : ι)
Goal: MeasurableSet (φ i)
-- MeasureTheory.AECover.ae_eventually_mem
structure AECover (φ : ι → Set α) : Prop where
  ae_eventually_mem : ∀ᵐ x ∂μ, ∀ᶠ i in l, x ∈ φ i
  protected measurableSet : ∀ i, MeasurableSet <| φ i
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ : ι → Set α} (self : MeasureTheory.AECover μ l φ)
Goal: ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in l, x ∈ φ i
-- MeasureTheory.AECover.inter
theorem inter {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hψ : AECover μ l ψ) :
    AECover μ l (fun i ↦ φ i ∩ ψ i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ ψ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hψ : MeasureTheory.AECover μ l ψ)
Goal: MeasureTheory.AECover μ l fun i => φ i ∩ ψ i
-- MeasureTheory.AECover.superset
theorem superset {φ ψ : ι → Set α} (hφ : AECover μ l φ) (hsub : ∀ i, φ i ⊆ ψ i)
    (hmeas : ∀ i, MeasurableSet (ψ i)) : AECover μ l ψ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {φ ψ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hsub : ∀ (i : ι), φ i ⊆ ψ i) (hmeas : ∀ (i : ι), MeasurableSet (ψ i))
Goal: MeasureTheory.AECover μ l ψ
-- MeasureTheory.AECover.mono_ac
theorem mono_ac {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≪ μ) :
    AECover ν l φ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {ν : MeasureTheory.Measure α} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hle : ν ≪ μ)
Goal: MeasureTheory.AECover ν l φ
-- MeasureTheory.AECover.mono
theorem mono {ν : Measure α} {φ : ι → Set α} (hφ : AECover μ l φ) (hle : ν ≤ μ) :
    AECover ν l φ 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} {ν : MeasureTheory.Measure α} {φ : ι → Set α} (hφ : MeasureTheory.AECover μ l φ) (hle : ν ≤ μ)
Goal: MeasureTheory.AECover ν l φ
-- MeasureTheory.aecover_ball
theorem aecover_ball {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
    AECover μ l (fun i ↦ Metric.ball x (r i)) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [PseudoMetricSpace α] [OpensMeasurableSpace α] {x : α} {r : ι → ℝ} (hr : Filter.Tendsto r l Filter.atTop)
Goal: MeasureTheory.AECover μ l fun i => Metric.ball x (r i)
-- MeasureTheory.aecover_closedBall
theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
    AECover μ l (fun i ↦ Metric.closedBall x (r i)) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [PseudoMetricSpace α] [OpensMeasurableSpace α] {x : α} {r : ι → ℝ} (hr : Filter.Tendsto r l Filter.atTop)
Goal: MeasureTheory.AECover μ l fun i => Metric.closedBall x (r i)
-- MeasureTheory.aecover_Ici
theorem aecover_Ici : AECover μ l fun i => Ici (a i) 
Arguments: {α : Type u_1} {ι : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {l : Filter ι} [Preorder α] [TopologicalSpace α] [OrderClosedTopology α] [OpensMeasurableSpace α] {a : ι → α} (ha : Filter.Tendsto a l Filter.atBot)
Goal: MeasureTheory.AECover μ l fun i => Set.Ici (a i)