ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intervalIntegral.continuousOn_primitive_interval_left
theorem continuousOn_primitive_interval_left (h_int : IntegrableOn f (uIcc a b) μ) :
    ContinuousOn (fun x => ∫ t in x..b, f t ∂μ) (uIcc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : MeasureTheory.IntegrableOn f (Set.uIcc a b) μ)
Goal: ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) (Set.uIcc a b)
-- intervalIntegral.continuous_primitive
theorem continuous_primitive (h_int : ∀ a b, IntervalIntegrable f μ a b) (a : ℝ) :
    Continuous fun b => ∫ x in a..b, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : ∀ (a b : ℝ), IntervalIntegrable f μ a b) (a : ℝ)
Goal: Continuous fun b => ∫ (x : ℝ) in a..b, f x ∂μ
-- MeasureTheory.Integrable.continuous_primitive
nonrec theorem _root_.MeasureTheory.Integrable.continuous_primitive (h_int : Integrable f μ)
    (a : ℝ) : Continuous fun b => ∫ x in a..b, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : MeasureTheory.Measure ℝ} {f : ℝ → E} [MeasureTheory.NoAtoms μ] (h_int : MeasureTheory.Integrable f μ) (a : ℝ)
Goal: Continuous fun b => ∫ (x : ℝ) in a..b, f x ∂μ
-- intervalIntegral.continuous_parametric_primitive_of_continuous
theorem continuous_parametric_primitive_of_continuous
    {a₀ : ℝ} (hf : Continuous f.uncurry) :
    Continuous fun p : X × ℝ ↦ ∫ t in a₀..p.2, f p.1 t ∂μ 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : X → ℝ → E} {a₀ : ℝ} (hf : Continuous (Function.uncurry f))
Goal: Continuous fun p => ∫ (t : ℝ) in a₀..p.2, f p.1 t ∂μ
-- intervalIntegral.continuous_parametric_intervalIntegral_of_continuous
@[fun_prop]
theorem continuous_parametric_intervalIntegral_of_continuous {a₀ : ℝ}
    (hf : Continuous f.uncurry) {s : X → ℝ} (hs : Continuous s) :
    Continuous fun x ↦ ∫ t in a₀..s x, f x t ∂μ 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : X → ℝ → E} {a₀ : ℝ} (hf : Continuous (Function.uncurry f)) {s : X → ℝ} (hs : Continuous s)
Goal: Continuous fun x => ∫ (t : ℝ) in a₀..s x, f x t ∂μ
-- intervalIntegral.continuous_parametric_intervalIntegral_of_continuous'
theorem continuous_parametric_intervalIntegral_of_continuous'
    (hf : Continuous f.uncurry) (a₀ b₀ : ℝ) :
    Continuous fun x ↦ ∫ t in a₀..b₀, f x t ∂μ 
Arguments: {E : Type u_1} {X : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [TopologicalSpace X] {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] [MeasureTheory.IsLocallyFiniteMeasure μ] {f : X → ℝ → E} (hf : Continuous (Function.uncurry f)) (a₀ b₀ : ℝ)
Goal: Continuous fun x => ∫ (t : ℝ) in a₀..b₀, f x t ∂μ
-- blimsup_cthickening_ae_le_of_eventually_mul_le_aux
theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
    (hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
    {M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
    (blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
      (blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] (p : ℕ → Prop) {s : ℕ → Set α} (hs : ∀ (i : ℕ), IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Filter.Tendsto r₁ Filter.atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁) {M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ (i : ℕ) in Filter.atTop, M * r₁ i ≤ r₂ i)
Goal: Filter.blimsup (fun i => Metric.cthickening (r₁ i) (s i)) Filter.atTop p ≤ᵐ[μ]
  Filter.blimsup (fun i => Metric.cthickening (r₂ i) (s i)) Filter.atTop p
-- blimsup_cthickening_ae_le_of_eventually_mul_le
theorem blimsup_cthickening_ae_le_of_eventually_mul_le (p : ℕ → Prop) {s : ℕ → Set α} {M : ℝ}
    (hM : 0 < M) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0))
    (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
    (blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
      (blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] (p : ℕ → Prop) {s : ℕ → Set α} {M : ℝ} (hM : 0 < M) {r₁ r₂ : ℕ → ℝ} (hr : Filter.Tendsto r₁ Filter.atTop (𝓝[>] 0)) (hMr : ∀ᶠ (i : ℕ) in Filter.atTop, M * r₁ i ≤ r₂ i)
Goal: Filter.blimsup (fun i => Metric.cthickening (r₁ i) (s i)) Filter.atTop p ≤ᵐ[μ]
  Filter.blimsup (fun i => Metric.cthickening (r₂ i) (s i)) Filter.atTop p
-- blimsup_cthickening_mul_ae_eq
theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
    (r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
    (blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
      (blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M) (r : ℕ → ℝ) (hr : Filter.Tendsto r Filter.atTop (𝓝 0))
Goal: Filter.blimsup (fun i => Metric.cthickening (M * r i) (s i)) Filter.atTop p =ᵐ[μ]
  Filter.blimsup (fun i => Metric.cthickening (r i) (s i)) Filter.atTop p
-- blimsup_cthickening_ae_eq_blimsup_thickening
theorem blimsup_cthickening_ae_eq_blimsup_thickening {p : ℕ → Prop} {s : ℕ → Set α} {r : ℕ → ℝ}
    (hr : Tendsto r atTop (𝓝 0)) (hr' : ∀ᶠ i in atTop, p i → 0 < r i) :
    (blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) =ᵐ[μ]
      (blimsup (fun i => thickening (r i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] {p : ℕ → Prop} {s : ℕ → Set α} {r : ℕ → ℝ} (hr : Filter.Tendsto r Filter.atTop (𝓝 0)) (hr' : ∀ᶠ (i : ℕ) in Filter.atTop, p i → 0 < r i)
Goal: Filter.blimsup (fun i => Metric.cthickening (r i) (s i)) Filter.atTop p =ᵐ[μ]
  Filter.blimsup (fun i => Metric.thickening (r i) (s i)) Filter.atTop p
-- blimsup_thickening_mul_ae_eq_aux
theorem blimsup_thickening_mul_ae_eq_aux (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
    (r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) (hr' : ∀ᶠ i in atTop, p i → 0 < r i) :
    (blimsup (fun i => thickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
      (blimsup (fun i => thickening (r i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M) (r : ℕ → ℝ) (hr : Filter.Tendsto r Filter.atTop (𝓝 0)) (hr' : ∀ᶠ (i : ℕ) in Filter.atTop, p i → 0 < r i)
Goal: Filter.blimsup (fun i => Metric.thickening (M * r i) (s i)) Filter.atTop p =ᵐ[μ]
  Filter.blimsup (fun i => Metric.thickening (r i) (s i)) Filter.atTop p
-- blimsup_thickening_mul_ae_eq
theorem blimsup_thickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M) (r : ℕ → ℝ)
    (hr : Tendsto r atTop (𝓝 0)) :
    (blimsup (fun i => thickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
      (blimsup (fun i => thickening (r i) (s i)) atTop p : Set α) 
Arguments: {α : Type u_1} [MetricSpace α] [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] [IsUnifLocDoublingMeasure μ] (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M) (r : ℕ → ℝ) (hr : Filter.Tendsto r Filter.atTop (𝓝 0))
Goal: Filter.blimsup (fun i => Metric.thickening (M * r i) (s i)) Filter.atTop p =ᵐ[μ]
  Filter.blimsup (fun i => Metric.thickening (r i) (s i)) Filter.atTop p
-- MeasureTheory.pow_mul_meas_ge_le_eLpNorm
theorem pow_mul_meas_ge_le_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
    (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ eLpNorm f p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) {f : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : MeasureTheory.AEStronglyMeasurable f μ) (ε : ℝ≥0∞)
Goal: (ε * μ {x | ε ≤ ↑‖f x‖₊ ^ p.toReal}) ^ (1 / p.toReal) ≤ MeasureTheory.eLpNorm f p μ
-- MeasureTheory.mul_meas_ge_le_pow_eLpNorm
theorem mul_meas_ge_le_pow_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
    ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal } ≤ eLpNorm f p μ ^ p.toReal 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) {f : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : MeasureTheory.AEStronglyMeasurable f μ) (ε : ℝ≥0∞)
Goal: ε * μ {x | ε ≤ ↑‖f x‖₊ ^ p.toReal} ≤ MeasureTheory.eLpNorm f p μ ^ p.toReal
-- MeasureTheory.mul_meas_ge_le_pow_eLpNorm'
theorem mul_meas_ge_le_pow_eLpNorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
    ε ^ p.toReal * μ { x | ε ≤ ‖f x‖₊ } ≤ eLpNorm f p μ ^ p.toReal 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) {f : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : MeasureTheory.AEStronglyMeasurable f μ) (ε : ℝ≥0∞)
Goal: ε ^ p.toReal * μ {x | ε ≤ ↑‖f x‖₊} ≤ MeasureTheory.eLpNorm f p μ ^ p.toReal
-- MeasureTheory.meas_ge_le_mul_pow_eLpNorm
theorem meas_ge_le_mul_pow_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
    (hf : AEStronglyMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
    μ { x | ε ≤ ‖f x‖₊ } ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal 
Arguments: {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} (μ : MeasureTheory.Measure α) {f : α → E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ⊤) (hf : MeasureTheory.AEStronglyMeasurable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0)
Goal: μ {x | ε ≤ ↑‖f x‖₊} ≤ ε⁻¹ ^ p.toReal * MeasureTheory.eLpNorm f p μ ^ p.toReal
-- Vitali.exists_disjoint_subfamily_covering_enlargment
theorem exists_disjoint_subfamily_covering_enlargment (B : ι → Set α) (t : Set ι) (δ : ι → ℝ)
    (τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R)
    (hne : ∀ a ∈ t, (B a).Nonempty) :
    ∃ u ⊆ t,
      u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b 
Arguments: {α : Type u_1} {ι : Type u_2} (B : ι → Set α) (t : Set ι) (δ : ι → ℝ) (τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R) (hne : ∀ a ∈ t, (B a).Nonempty)
Goal: ∃ u ⊆ t, u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b
-- Vitali.exists_disjoint_subfamily_covering_enlargment_closedBall
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) (τ : ℝ) (hτ : 3 < τ) :
    ∃ u ⊆ t,
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b) 
Arguments: {α : Type u_1} {ι : Type u_2} [MetricSpace α] (t : Set ι) (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) (τ : ℝ) (hτ : 3 < τ)
Goal: ∃ u ⊆ t,
  (u.PairwiseDisjoint fun a => Metric.closedBall (x a) (r a)) ∧
    ∀ a ∈ t, ∃ b ∈ u, Metric.closedBall (x a) (r a) ⊆ Metric.closedBall (x b) (τ * r b)
-- Vitali.exists_disjoint_covering_ae
theorem exists_disjoint_covering_ae [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
    [SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
    (C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
    (μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
    (ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
    (hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
    ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0 
Arguments: {α : Type u_1} {ι : Type u_2} [MetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [SecondCountableTopology α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι) (C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ Metric.closedBall (c a) (r a)) (μB : ∀ a ∈ t, μ (Metric.closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)) (ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a)) (hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x)
Goal: ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
-- MeasureTheory.StronglyMeasurable.inner
protected theorem inner {𝕜 : Type*} {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
    [InnerProductSpace 𝕜 E] {_ : MeasurableSpace α} {f g : α → E} (hf : StronglyMeasurable f)
    (hg : StronglyMeasurable g) : StronglyMeasurable fun t => @inner 𝕜 _ _ (f t) (g t) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ∀ {x : MeasurableSpace α} {f g : α → E},
  MeasureTheory.StronglyMeasurable f →
    MeasureTheory.StronglyMeasurable g → MeasureTheory.StronglyMeasurable fun t => ⟪f t, g t⟫_𝕜
-- MeasureTheory.AEStronglyMeasurable.re
protected theorem re {f : α → 𝕜} (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => RCLike.re (f x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => RCLike.re (f x)) μ
-- MeasureTheory.AEStronglyMeasurable.im
protected theorem im {f : α → 𝕜} (hf : AEStronglyMeasurable f μ) :
    AEStronglyMeasurable (fun x => RCLike.im (f x)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {𝕜 : Type u_2} [RCLike 𝕜] {f : α → 𝕜} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun x => RCLike.im (f x)) μ
-- MeasureTheory.AEStronglyMeasurable.inner
protected theorem inner {_ : MeasurableSpace α} {μ : Measure α} {f g : α → E}
    (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
    AEStronglyMeasurable (fun x => ⟪f x, g x⟫) μ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} {E : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
Goal: ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : α → E},
  MeasureTheory.AEStronglyMeasurable f μ →
    MeasureTheory.AEStronglyMeasurable g μ → MeasureTheory.AEStronglyMeasurable (fun x => ⟪f x, g x⟫_𝕜) μ
-- MeasureTheory.eLpNorm'_add_le
theorem eLpNorm'_add_le (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hq1 : 1 ≤ q) : eLpNorm' (f + g) q μ ≤ eLpNorm' f q μ + eLpNorm' g q μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {q : ℝ} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hq1 : 1 ≤ q)
Goal: MeasureTheory.eLpNorm' (f + g) q μ ≤ MeasureTheory.eLpNorm' f q μ + MeasureTheory.eLpNorm' g q μ
-- MeasureTheory.eLpNorm'_add_le_of_le_one
theorem eLpNorm'_add_le_of_le_one (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤ q) (hq1 : q ≤ 1) :
    eLpNorm' (f + g) q μ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (eLpNorm' f q μ + eLpNorm' g q μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {q : ℝ} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hq0 : 0 ≤ q) (hq1 : q ≤ 1)
Goal: MeasureTheory.eLpNorm' (f + g) q μ ≤ 2 ^ (1 / q - 1) * (MeasureTheory.eLpNorm' f q μ + MeasureTheory.eLpNorm' g q μ)
-- MeasureTheory.eLpNormEssSup_add_le
theorem eLpNormEssSup_add_le {f g : α → E} :
    eLpNormEssSup (f + g) μ ≤ eLpNormEssSup f μ + eLpNormEssSup g μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f g : α → E}
Goal: MeasureTheory.eLpNormEssSup (f + g) μ ≤ MeasureTheory.eLpNormEssSup f μ + MeasureTheory.eLpNormEssSup g μ
-- MeasureTheory.eLpNorm_add_le
theorem eLpNorm_add_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hp1 : 1 ≤ p) : eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hp1 : 1 ≤ p)
Goal: MeasureTheory.eLpNorm (f + g) p μ ≤ MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ
-- MeasureTheory.LpAddConst_of_one_le
theorem LpAddConst_of_one_le {p : ℝ≥0∞} (hp : 1 ≤ p) : LpAddConst p = 1 
Arguments: {p : ℝ≥0∞} (hp : 1 ≤ p)
Goal: MeasureTheory.LpAddConst p = 1
-- MeasureTheory.LpAddConst_zero
theorem LpAddConst_zero : LpAddConst 0 = 1 
Goal: MeasureTheory.LpAddConst 0 = 1
-- MeasureTheory.LpAddConst_lt_top
theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ 
Arguments: (p : ℝ≥0∞)
Goal: MeasureTheory.LpAddConst p < ⊤
-- MeasureTheory.eLpNorm_add_le'
theorem eLpNorm_add_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (p : ℝ≥0∞) : eLpNorm (f + g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm (f + g) p μ ≤
  MeasureTheory.LpAddConst p * (MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ)
-- MeasureTheory.exists_Lp_half
theorem exists_Lp_half (p : ℝ≥0∞) {δ : ℝ≥0∞} (hδ : δ ≠ 0) :
    ∃ η : ℝ≥0∞,
      0 < η ∧
        ∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ →
          eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < δ 
Arguments: {α : Type u_1} (E : Type u_2) {m : MeasurableSpace α} [NormedAddCommGroup E] (μ : MeasureTheory.Measure α) (p : ℝ≥0∞) {δ : ℝ≥0∞} (hδ : δ ≠ 0)
Goal: ∃ η,
  0 < η ∧
    ∀ (f g : α → E),
      MeasureTheory.AEStronglyMeasurable f μ →
        MeasureTheory.AEStronglyMeasurable g μ →
          MeasureTheory.eLpNorm f p μ ≤ η → MeasureTheory.eLpNorm g p μ ≤ η → MeasureTheory.eLpNorm (f + g) p μ < δ
-- MeasureTheory.eLpNorm_sub_le'
theorem eLpNorm_sub_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (p : ℝ≥0∞) : eLpNorm (f - g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (p : ℝ≥0∞)
Goal: MeasureTheory.eLpNorm (f - g) p μ ≤
  MeasureTheory.LpAddConst p * (MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ)
-- MeasureTheory.eLpNorm_sub_le
theorem eLpNorm_sub_le {f g : α → E} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
    (hp : 1 ≤ p) : eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.AEStronglyMeasurable f μ) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hp : 1 ≤ p)
Goal: MeasureTheory.eLpNorm (f - g) p μ ≤ MeasureTheory.eLpNorm f p μ + MeasureTheory.eLpNorm g p μ
-- MeasureTheory.eLpNorm_add_lt_top
theorem eLpNorm_add_lt_top {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) :
    eLpNorm (f + g) p μ < ∞ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.eLpNorm (f + g) p μ < ⊤
-- MeasureTheory.eLpNorm'_sum_le
theorem eLpNorm'_sum_le {ι} {f : ι → α → E} {s : Finset ι}
    (hfs : ∀ i, i ∈ s → AEStronglyMeasurable (f i) μ) (hq1 : 1 ≤ q) :
    eLpNorm' (∑ i ∈ s, f i) q μ ≤ ∑ i ∈ s, eLpNorm' (f i) q μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {q : ℝ} {μ : MeasureTheory.Measure α} {ι : Type u_3} {f : ι → α → E} {s : Finset ι} (hfs : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ) (hq1 : 1 ≤ q)
Goal: MeasureTheory.eLpNorm' (∑ i ∈ s, f i) q μ ≤ ∑ i ∈ s, MeasureTheory.eLpNorm' (f i) q μ
-- MeasureTheory.eLpNorm_sum_le
theorem eLpNorm_sum_le {ι} {f : ι → α → E} {s : Finset ι}
    (hfs : ∀ i, i ∈ s → AEStronglyMeasurable (f i) μ) (hp1 : 1 ≤ p) :
    eLpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, eLpNorm (f i) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {ι : Type u_3} {f : ι → α → E} {s : Finset ι} (hfs : ∀ i ∈ s, MeasureTheory.AEStronglyMeasurable (f i) μ) (hp1 : 1 ≤ p)
Goal: MeasureTheory.eLpNorm (∑ i ∈ s, f i) p μ ≤ ∑ i ∈ s, MeasureTheory.eLpNorm (f i) p μ
-- MeasureTheory.Memℒp.add
theorem Memℒp.add {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : Memℒp (f + g) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp (f + g) p μ
-- MeasureTheory.Memℒp.sub
theorem Memℒp.sub {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : Memℒp (f - g) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {f g : α → E} (hf : MeasureTheory.Memℒp f p μ) (hg : MeasureTheory.Memℒp g p μ)
Goal: MeasureTheory.Memℒp (f - g) p μ
-- MeasureTheory.memℒp_finset_sum
theorem memℒp_finset_sum {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
    Memℒp (fun a => ∑ i ∈ s, f i a) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {ι : Type u_3} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.Memℒp (fun a => ∑ i ∈ s, f i a) p μ
-- MeasureTheory.memℒp_finset_sum'
theorem memℒp_finset_sum' {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
    Memℒp (∑ i ∈ s, f i) p μ 
Arguments: {α : Type u_1} {E : Type u_2} {m : MeasurableSpace α} [NormedAddCommGroup E] {p : ℝ≥0∞} {μ : MeasureTheory.Measure α} {ι : Type u_3} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, MeasureTheory.Memℒp (f i) p μ)
Goal: MeasureTheory.Memℒp (∑ i ∈ s, f i) p μ
-- DomMulAct.smul_Lp_val
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [SMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] [MeasurableSMul M α] (c : Mᵈᵐᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑(c • f) = c • ↑f
-- DomAddAct.vadd_Lp_val
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 
Arguments: {M : Type u_1} {α : Type u_3} {E : Type u_4} [MeasurableSpace M] [MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞} [VAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] [MeasurableVAdd M α] (c : Mᵈᵃᵃ) (f : ↥(MeasureTheory.Lp E p μ))
Goal: ↑(c +ᵥ f) = c +ᵥ ↑f