ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- essSup_measure_zero
@[simp]
theorem essSup_measure_zero {m : MeasurableSpace α} {f : α → β} : essSup f (0 : Measure α) = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice β] {m : MeasurableSpace α} {f : α → β}
Goal: essSup f 0 = ⊥
-- essInf_measure_zero
@[simp]
theorem essInf_measure_zero {_ : MeasurableSpace α} {f : α → β} : essInf f (0 : Measure α) = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice β]
Goal: ∀ {x : MeasurableSpace α} {f : α → β}, essInf f 0 = ⊤
-- essSup_mono_ae
theorem essSup_mono_ae {f g : α → β} (hfg : f ≤ᵐ[μ] g) : essSup f μ ≤ essSup g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {f g : α → β} (hfg : f ≤ᵐ[μ] g)
Goal: essSup f μ ≤ essSup g μ
-- essInf_mono_ae
theorem essInf_mono_ae {f g : α → β} (hfg : f ≤ᵐ[μ] g) : essInf f μ ≤ essInf g μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {f g : α → β} (hfg : f ≤ᵐ[μ] g)
Goal: essInf f μ ≤ essInf g μ
-- essSup_le_of_ae_le
theorem essSup_le_of_ae_le {f : α → β} (c : β) (hf : f ≤ᵐ[μ] fun _ => c) : essSup f μ ≤ c 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (c : β) (hf : f ≤ᵐ[μ] fun x => c)
Goal: essSup f μ ≤ c
-- le_essInf_of_ae_le
theorem le_essInf_of_ae_le {f : α → β} (c : β) (hf : (fun _ => c) ≤ᵐ[μ] f) : c ≤ essInf f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (c : β) (hf : (fun x => c) ≤ᵐ[μ] f)
Goal: c ≤ essInf f μ
-- essSup_const_bot
theorem essSup_const_bot : essSup (fun _ : α => (⊥ : β)) μ = (⊥ : β) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β]
Goal: essSup (fun x => ⊥) μ = ⊥
-- essInf_const_top
theorem essInf_const_top : essInf (fun _ : α => (⊤ : β)) μ = (⊤ : β) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β]
Goal: essInf (fun x => ⊤) μ = ⊤
-- OrderIso.essSup_apply
theorem OrderIso.essSup_apply {m : MeasurableSpace α} {γ} [CompleteLattice γ] (f : α → β)
    (μ : Measure α) (g : β ≃o γ) : g (essSup f μ) = essSup (fun x => g (f x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice β] {m : MeasurableSpace α} {γ : Type u_3} [CompleteLattice γ] (f : α → β) (μ : MeasureTheory.Measure α) (g : β ≃o γ)
Goal: g (essSup f μ) = essSup (fun x => g (f x)) μ
-- OrderIso.essInf_apply
theorem OrderIso.essInf_apply {_ : MeasurableSpace α} {γ} [CompleteLattice γ] (f : α → β)
    (μ : Measure α) (g : β ≃o γ) : g (essInf f μ) = essInf (fun x => g (f x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice β]
Goal: ∀ {x : MeasurableSpace α} {γ : Type u_3} [inst : CompleteLattice γ] (f : α → β) (μ : MeasureTheory.Measure α)
  (g : β ≃o γ), g (essInf f μ) = essInf (fun x => g (f x)) μ
-- essSup_mono_measure
theorem essSup_mono_measure {f : α → β} (hμν : ν ≪ μ) : essSup f ν ≤ essSup f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (hμν : ν ≪ μ)
Goal: essSup f ν ≤ essSup f μ
-- essSup_mono_measure'
theorem essSup_mono_measure' {α : Type*} {β : Type*} {_ : MeasurableSpace α}
    {μ ν : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (hμν : ν ≤ μ) :
    essSup f ν ≤ essSup f μ 
Arguments: {α : Type u_3} {β : Type u_4}
Goal: ∀ {x : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [inst : CompleteLattice β] {f : α → β},
  ν ≤ μ → essSup f ν ≤ essSup f μ
-- essInf_antitone_measure
theorem essInf_antitone_measure {f : α → β} (hμν : μ ≪ ν) : essInf f ν ≤ essInf f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} (hμν : μ ≪ ν)
Goal: essInf f ν ≤ essInf f μ
-- essSup_smul_measure
theorem essSup_smul_measure {f : α → β} {c : ℝ≥0∞} (hc : c ≠ 0) :
    essSup f (c • μ) = essSup f μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {f : α → β} {c : ℝ≥0∞} (hc : c ≠ 0)
Goal: essSup f (c • μ) = essSup f μ
-- essSup_comp_le_essSup_map_measure
theorem essSup_comp_le_essSup_map_measure (hf : AEMeasurable f μ) :
    essSup (g ∘ f) μ ≤ essSup g (Measure.map f μ) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} (hf : AEMeasurable f μ)
Goal: essSup (g ∘ f) μ ≤ essSup g (MeasureTheory.Measure.map f μ)
-- MeasurableEmbedding.essSup_map_measure
theorem MeasurableEmbedding.essSup_map_measure (hf : MeasurableEmbedding f) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} (hf : MeasurableEmbedding f)
Goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ
-- essSup_map_measure_of_measurable
theorem essSup_map_measure_of_measurable (hg : Measurable g) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} [MeasurableSpace β] [TopologicalSpace β] [SecondCountableTopology β] [OrderClosedTopology β] [OpensMeasurableSpace β] (hg : Measurable g) (hf : AEMeasurable f μ)
Goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ
-- essSup_map_measure
theorem essSup_map_measure (hg : AEMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
    essSup g (Measure.map f μ) = essSup (g ∘ f) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [CompleteLattice β] {γ : Type u_3} {mγ : MeasurableSpace γ} {f : α → γ} {g : γ → β} [MeasurableSpace β] [TopologicalSpace β] [SecondCountableTopology β] [OrderClosedTopology β] [OpensMeasurableSpace β] (hg : AEMeasurable g (MeasureTheory.Measure.map f μ)) (hf : AEMeasurable f μ)
Goal: essSup g (MeasureTheory.Measure.map f μ) = essSup (g ∘ f) μ
-- ENNReal.essSup_piecewise
lemma essSup_piecewise {s : Set α} [DecidablePred (· ∈ s)] {g} (hs : MeasurableSet s) :
    essSup (s.piecewise f g) μ = max (essSup f (μ.restrict s)) (essSup g (μ.restrict sᶜ)) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {s : Set α} [DecidablePred fun x => x ∈ s] {g : α → ℝ≥0∞} (hs : MeasurableSet s)
Goal: essSup (s.piecewise f g) μ = max (essSup f (μ.restrict s)) (essSup g (μ.restrict sᶜ))
-- ENNReal.essSup_indicator_eq_essSup_restrict
theorem essSup_indicator_eq_essSup_restrict {s : Set α} {f : α → ℝ≥0∞} (hs : MeasurableSet s) :
    essSup (s.indicator f) μ = essSup f (μ.restrict s) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hs : MeasurableSet s)
Goal: essSup (s.indicator f) μ = essSup f (μ.restrict s)
-- ENNReal.ae_le_essSup
theorem ae_le_essSup (f : α → ℝ≥0∞) : ∀ᵐ y ∂μ, f y ≤ essSup f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f : α → ℝ≥0∞)
Goal: ∀ᵐ (y : α) ∂μ, f y ≤ essSup f μ
-- ENNReal.essSup_eq_zero_iff
@[simp]
theorem essSup_eq_zero_iff : essSup f μ = 0 ↔ f =ᵐ[μ] 0 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞}
Goal: essSup f μ = 0 ↔ f =ᵐ[μ] 0
-- ENNReal.essSup_const_mul
theorem essSup_const_mul {a : ℝ≥0∞} : essSup (fun x : α => a * f x) μ = a * essSup f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0∞} {a : ℝ≥0∞}
Goal: essSup (fun x => a * f x) μ = a * essSup f μ
-- ENNReal.essSup_mul_le
theorem essSup_mul_le (f g : α → ℝ≥0∞) : essSup (f * g) μ ≤ essSup f μ * essSup g μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ≥0∞)
Goal: essSup (f * g) μ ≤ essSup f μ * essSup g μ
-- ENNReal.essSup_add_le
theorem essSup_add_le (f g : α → ℝ≥0∞) : essSup (f + g) μ ≤ essSup f μ + essSup g μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} (f g : α → ℝ≥0∞)
Goal: essSup (f + g) μ ≤ essSup f μ + essSup g μ
-- ENNReal.essSup_liminf_le
theorem essSup_liminf_le {ι} [Countable ι] [LinearOrder ι] (f : ι → α → ℝ≥0∞) :
    essSup (fun x => atTop.liminf fun n => f n x) μ ≤
      atTop.liminf fun n => essSup (fun x => f n x) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} [Countable ι] [LinearOrder ι] (f : ι → α → ℝ≥0∞)
Goal: essSup (fun x => Filter.liminf (fun n => f n x) Filter.atTop) μ ≤
  Filter.liminf (fun n => essSup (fun x => f n x) μ) Filter.atTop
-- ENNReal.coe_essSup
theorem coe_essSup {f : α → ℝ≥0} (hf : IsBoundedUnder (· ≤ ·) (ae μ) f) :
    ((essSup f μ : ℝ≥0) : ℝ≥0∞) = essSup (fun x => (f x : ℝ≥0∞)) μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ≥0} (hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (MeasureTheory.ae μ) f)
Goal: ↑(essSup f μ) = essSup (fun x => ↑(f x)) μ
-- ENNReal.essSup_restrict_eq_of_support_subset
lemma essSup_restrict_eq_of_support_subset {s : Set α} {f : α → ℝ≥0∞} (hsf : f.support ⊆ s) :
    essSup f (μ.restrict s) = essSup f μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} {f : α → ℝ≥0∞} (hsf : Function.support f ⊆ s)
Goal: essSup f (μ.restrict s) = essSup f μ
-- ENNReal.lintegral_mul_le_one_of_lintegral_rpow_eq_one
theorem lintegral_mul_le_one_of_lintegral_rpow_eq_one {p q : ℝ} (hpq : p.IsConjExponent q)
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_norm : ∫⁻ a, f a ^ p ∂μ = 1)
    (hg_norm : ∫⁻ a, g a ^ q ∂μ = 1) : (∫⁻ a, (f * g) a ∂μ) ≤ 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1) (hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1)
Goal: ∫⁻ (a : α), (f * g) a ∂μ ≤ 1
-- ENNReal.fun_eq_funMulInvSnorm_mul_eLpNorm
theorem fun_eq_funMulInvSnorm_mul_eLpNorm {p : ℝ} (f : α → ℝ≥0∞)
    (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) {a : α} :
    f a = funMulInvSnorm f p μ a * (∫⁻ c, f c ^ p ∂μ) ^ (1 / p) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} (f : α → ENNReal) (hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤) {a : α}
Goal: f a = ENNReal.funMulInvSnorm f p μ a * (∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)
-- ENNReal.funMulInvSnorm_rpow
theorem funMulInvSnorm_rpow {p : ℝ} (hp0 : 0 < p) {f : α → ℝ≥0∞} {a : α} :
    funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ c, f c ^ p ∂μ)⁻¹ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} (hp0 : 0 < p) {f : α → ENNReal} {a : α}
Goal: ENNReal.funMulInvSnorm f p μ a ^ p = f a ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹
-- ENNReal.lintegral_rpow_funMulInvSnorm_eq_one
theorem lintegral_rpow_funMulInvSnorm_eq_one {p : ℝ} (hp0_lt : 0 < p) {f : α → ℝ≥0∞}
    (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) :
    ∫⁻ c, funMulInvSnorm f p μ c ^ p ∂μ = 1 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} (hp0_lt : 0 < p) {f : α → ENNReal} (hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤)
Goal: ∫⁻ (c : α), ENNReal.funMulInvSnorm f p μ c ^ p ∂μ = 1
-- ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top
theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_ne_top {p q : ℝ} (hpq : p.IsConjExponent q)
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_nontop : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤)
    (hg_nontop : (∫⁻ a, g a ^ q ∂μ) ≠ ⊤) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
    (hg_nonzero : (∫⁻ a, g a ^ q ∂μ) ≠ 0) :
    (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤) (hg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ⊤) (hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0) (hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0)
Goal: ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
-- ENNReal.ae_eq_zero_of_lintegral_rpow_eq_zero
theorem ae_eq_zero_of_lintegral_rpow_eq_zero {p : ℝ} (hp0 : 0 ≤ p) {f : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_zero : ∫⁻ a, f a ^ p ∂μ = 0) : f =ᵐ[μ] 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} (hp0 : 0 ≤ p) {f : α → ENNReal} (hf : AEMeasurable f μ) (hf_zero : ∫⁻ (a : α), f a ^ p ∂μ = 0)
Goal: f =ᵐ[μ] 0
-- ENNReal.lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero
theorem lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero {p : ℝ} (hp0 : 0 ≤ p) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_zero : ∫⁻ a, f a ^ p ∂μ = 0) : (∫⁻ a, (f * g) a ∂μ) = 0 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} (hp0 : 0 ≤ p) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_zero : ∫⁻ (a : α), f a ^ p ∂μ = 0)
Goal: ∫⁻ (a : α), (f * g) a ∂μ = 0
-- ENNReal.lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top
theorem lintegral_mul_le_Lp_mul_Lq_of_ne_zero_of_eq_top {p q : ℝ} (hp0_lt : 0 < p) (hq0 : 0 ≤ q)
    {f g : α → ℝ≥0∞} (hf_top : ∫⁻ a, f a ^ p ∂μ = ⊤) (hg_nonzero : (∫⁻ a, g a ^ q ∂μ) ≠ 0) :
    (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hp0_lt : 0 < p) (hq0 : 0 ≤ q) {f g : α → ENNReal} (hf_top : ∫⁻ (a : α), f a ^ p ∂μ = ⊤) (hg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0)
Goal: ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
-- ENNReal.lintegral_mul_le_Lp_mul_Lq
theorem lintegral_mul_le_Lp_mul_Lq (μ : Measure α) {p q : ℝ} (hpq : p.IsConjExponent q)
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    (∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: ∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)
-- ENNReal.lintegral_mul_norm_pow_le
theorem lintegral_mul_norm_pow_le {α} [MeasurableSpace α] {μ : Measure α}
    {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
    {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) (hpq : p + q = 1) :
    ∫⁻ a, f a ^ p * g a ^ q ∂μ ≤ (∫⁻ a, f a ∂μ) ^ p * (∫⁻ a, g a ∂μ) ^ q 
Arguments: {α : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f g : α → ENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) (hpq : p + q = 1)
Goal: ∫⁻ (a : α), f a ^ p * g a ^ q ∂μ ≤ (∫⁻ (a : α), f a ∂μ) ^ p * (∫⁻ (a : α), g a ∂μ) ^ q
-- ENNReal.lintegral_prod_norm_pow_le
theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
    (s : Finset ι) {f : ι → α → ℝ≥0∞} (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
    {p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) :
    ∫⁻ a, ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i 
Arguments: {α : Type u_2} {ι : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Finset ι) {f : ι → α → ENNReal} (hf : ∀ i ∈ s, AEMeasurable (f i) μ) {p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i)
Goal: ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
-- ENNReal.lintegral_mul_prod_norm_pow_le
theorem lintegral_mul_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
    (s : Finset ι) {g : α →  ℝ≥0∞} {f : ι → α → ℝ≥0∞} (hg : AEMeasurable g μ)
    (hf : ∀ i ∈ s, AEMeasurable (f i) μ) (q : ℝ) {p : ι → ℝ} (hpq : q + ∑ i ∈ s, p i = 1)
    (hq :  0 ≤ q) (hp : ∀ i ∈ s, 0 ≤ p i) :
    ∫⁻ a, g a ^ q * ∏ i ∈ s, f i a ^ p i ∂μ ≤
      (∫⁻ a, g a ∂μ) ^ q * ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i 
Arguments: {α : Type u_2} {ι : Type u_3} [MeasurableSpace α] {μ : MeasureTheory.Measure α} (s : Finset ι) {g : α → ENNReal} {f : ι → α → ENNReal} (hg : AEMeasurable g μ) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) (q : ℝ) {p : ι → ℝ} (hpq : q + ∑ i ∈ s, p i = 1) (hq : 0 ≤ q) (hp : ∀ i ∈ s, 0 ≤ p i)
Goal: ∫⁻ (a : α), g a ^ q * ∏ i ∈ s, f i a ^ p i ∂μ ≤ (∫⁻ (a : α), g a ∂μ) ^ q * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
-- ENNReal.lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top
theorem lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top {p : ℝ} {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) < ⊤) (hg_top : (∫⁻ a, g a ^ p ∂μ) < ⊤)
    (hp1 : 1 ≤ p) : (∫⁻ a, (f + g) a ^ p ∂μ) < ⊤ 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ⊤) (hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ⊤) (hp1 : 1 ≤ p)
Goal: ∫⁻ (a : α), (f + g) a ^ p ∂μ < ⊤
-- ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr
theorem lintegral_Lp_mul_le_Lq_mul_Lr {α} [MeasurableSpace α] {p q r : ℝ} (hp0_lt : 0 < p)
    (hpq : p < q) (hpqr : 1 / p = 1 / q + 1 / r) (μ : Measure α) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    (∫⁻ a, (f * g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ q ∂μ) ^ (1 / q) * (∫⁻ a, g a ^ r ∂μ) ^ (1 / r) 
Arguments: {α : Type u_2} [MeasurableSpace α] {p q r : ℝ} (hp0_lt : 0 < p) (hpq : p < q) (hpqr : 1 / p = 1 / q + 1 / r) (μ : MeasureTheory.Measure α) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ^ (1 / q) * (∫⁻ (a : α), g a ^ r ∂μ) ^ (1 / r)
-- ENNReal.lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow
theorem lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow {p q : ℝ}
    (hpq : p.IsConjExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, f a * g a ^ (p - 1) ∂μ) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ p ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤)
Goal: ∫⁻ (a : α), f a * g a ^ (p - 1) ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q)
-- ENNReal.lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add
theorem lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤)
Goal: ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤
  ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)
-- _private.Mathlib.MeasureTheory.Integral.MeanInequalities.0.ENNReal.lintegral_Lp_add_le_aux
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ENNReal} (hf : AEMeasurable f μ) (hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤) (h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0) (h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤)
Goal: (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
-- ENNReal.lintegral_Lp_add_le
theorem lintegral_Lp_add_le {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
    (hp1 : 1 ≤ p) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} {f g : α → ENNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hp1 : 1 ≤ p)
Goal: (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
-- ENNReal.lintegral_Lp_add_le_of_le_one
theorem lintegral_Lp_add_le_of_le_one {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hp0 : 0 ≤ p)
    (hp1 : p ≤ 1) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (2 : ℝ≥0∞) ^ (1 / p - 1) * ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p : ℝ} {f g : α → ENNReal} (hf : AEMeasurable f μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
Goal: (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤
  2 ^ (1 / p - 1) * ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
-- NNReal.lintegral_mul_le_Lp_mul_Lq
theorem NNReal.lintegral_mul_le_Lp_mul_Lq {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ≥0}
    (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    (∫⁻ a, (f * g) a ∂μ) ≤
      (∫⁻ a, (f a : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) * (∫⁻ a, (g a : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → NNReal} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: ∫⁻ (a : α), ↑((f * g) a) ∂μ ≤ (∫⁻ (a : α), ↑(f a) ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), ↑(g a) ^ q ∂μ) ^ (1 / q)
-- MeasurableSpace.isPiSystem_measurableSet
theorem isPiSystem_measurableSet {α : Type*} [MeasurableSpace α] :
    IsPiSystem { s : Set α | MeasurableSet s } 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: IsPiSystem {s | MeasurableSet s}
-- IsPiSystem.singleton
theorem IsPiSystem.singleton {α} (S : Set α) : IsPiSystem ({S} : Set (Set α)) 
Arguments: {α : Type u_1} (S : Set α)
Goal: IsPiSystem {S}
-- IsPiSystem.insert_empty
theorem IsPiSystem.insert_empty {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
    IsPiSystem (insert ∅ S) 
Arguments: {α : Type u_1} {S : Set (Set α)} (h_pi : IsPiSystem S)
Goal: IsPiSystem (insert ∅ S)
-- IsPiSystem.insert_univ
theorem IsPiSystem.insert_univ {α} {S : Set (Set α)} (h_pi : IsPiSystem S) :
    IsPiSystem (insert Set.univ S) 
Arguments: {α : Type u_1} {S : Set (Set α)} (h_pi : IsPiSystem S)
Goal: IsPiSystem (insert Set.univ S)
-- IsPiSystem.comap
theorem IsPiSystem.comap {α β} {S : Set (Set β)} (h_pi : IsPiSystem S) (f : α → β) :
    IsPiSystem { s : Set α | ∃ t ∈ S, f ⁻¹' t = s } 
Arguments: {α : Type u_1} {β : Type u_2} {S : Set (Set β)} (h_pi : IsPiSystem S) (f : α → β)
Goal: IsPiSystem {s | ∃ t ∈ S, f ⁻¹' t = s}
-- isPiSystem_iUnion_of_directed_le
theorem isPiSystem_iUnion_of_directed_le {α ι} (p : ι → Set (Set α))
    (hp_pi : ∀ n, IsPiSystem (p n)) (hp_directed : Directed (· ≤ ·) p) :
    IsPiSystem (⋃ n, p n) 
Arguments: {α : Type u_1} {ι : Sort u_2} (p : ι → Set (Set α)) (hp_pi : ∀ (n : ι), IsPiSystem (p n)) (hp_directed : Directed (fun x x_1 => x ≤ x_1) p)
Goal: IsPiSystem (⋃ n, p n)
-- isPiSystem_iUnion_of_monotone
theorem isPiSystem_iUnion_of_monotone {α ι} [SemilatticeSup ι] (p : ι → Set (Set α))
    (hp_pi : ∀ n, IsPiSystem (p n)) (hp_mono : Monotone p) : IsPiSystem (⋃ n, p n) 
Arguments: {α : Type u_1} {ι : Type u_2} [SemilatticeSup ι] (p : ι → Set (Set α)) (hp_pi : ∀ (n : ι), IsPiSystem (p n)) (hp_mono : Monotone p)
Goal: IsPiSystem (⋃ n, p n)
-- isPiSystem_image_Iio
theorem isPiSystem_image_Iio (s : Set α) : IsPiSystem (Iio '' s) 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α)
Goal: IsPiSystem (Set.Iio '' s)