ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ClosedEmbedding.integral_map
theorem _root_.ClosedEmbedding.integral_map {β} [TopologicalSpace α] [BorelSpace α]
    [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {φ : α → β} (hφ : ClosedEmbedding φ)
    (f : β → G) : ∫ y, f y ∂Measure.map φ μ = ∫ x, f (φ x) ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [TopologicalSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β] {φ : α → β} (hφ : ClosedEmbedding φ) (f : β → G)
Goal: ∫ (y : β), f y ∂MeasureTheory.Measure.map φ μ = ∫ (x : α), f (φ x) ∂μ
-- MeasureTheory.integral_map_equiv
theorem integral_map_equiv {β} [MeasurableSpace β] (e : α ≃ᵐ β) (f : β → G) :
    ∫ y, f y ∂Measure.map e μ = ∫ x, f (e x) ∂μ 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [MeasurableSpace β] (e : α ≃ᵐ β) (f : β → G)
Goal: ∫ (y : β), f y ∂MeasureTheory.Measure.map (⇑e) μ = ∫ (x : α), f (e x) ∂μ
-- MeasureTheory.MeasurePreserving.integral_comp
theorem MeasurePreserving.integral_comp {β} {_ : MeasurableSpace β} {f : α → β} {ν}
    (h₁ : MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) (g : β → G) :
    ∫ x, g (f x) ∂μ = ∫ y, g y ∂ν 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7}
Goal: ∀ {x : MeasurableSpace β} {f : α → β} {ν : MeasureTheory.Measure β},
  MeasureTheory.MeasurePreserving f μ ν →
    MeasurableEmbedding f → ∀ (g : β → G), ∫ (x : α), g (f x) ∂μ = ∫ (y : β), g y ∂ν
-- MeasureTheory.MeasurePreserving.integral_comp'
theorem MeasurePreserving.integral_comp' {β} [MeasurableSpace β] {ν} {f : α ≃ᵐ β}
    (h : MeasurePreserving f μ ν) (g : β → G) :
    ∫ x, g (f x) ∂μ = ∫ y, g y ∂ν 
Arguments: {α : Type u_1} {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_7} [MeasurableSpace β] {ν : MeasureTheory.Measure β} {f : α ≃ᵐ β} (h : MeasureTheory.MeasurePreserving (⇑f) μ ν) (g : β → G)
Goal: ∫ (x : α), g (f x) ∂μ = ∫ (y : β), g y ∂ν
-- MeasureTheory.integral_subtype_comap
theorem integral_subtype_comap {α} [MeasurableSpace α] {μ : Measure α} {s : Set α}
    (hs : MeasurableSet s) (f : α → G) :
    ∫ x : s, f (x : α) ∂(Measure.comap Subtype.val μ) = ∫ x in s, f x ∂μ 
Arguments: {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {α : Type u_7} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : MeasurableSet s) (f : α → G)
Goal: ∫ (x : ↑s), f ↑x ∂MeasureTheory.Measure.comap Subtype.val μ = ∫ (x : α) in s, f x ∂μ
-- MeasureTheory.integral_subtype
theorem integral_subtype {α} [MeasureSpace α] {s : Set α} (hs : MeasurableSet s) (f : α → G) :
    ∫ x : s, f x = ∫ x in s, f x 
Arguments: {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {α : Type u_7} [MeasureTheory.MeasureSpace α] {s : Set α} (hs : MeasurableSet s) (f : α → G)
Goal: ∫ (x : ↑s), f ↑x = ∫ (x : α) in s, f x
-- MeasureTheory.integral_dirac'
@[simp]
theorem integral_dirac' [MeasurableSpace α] (f : α → E) (a : α) (hfm : StronglyMeasurable f) :
    ∫ x, f x ∂Measure.dirac a = f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] [MeasurableSpace α] (f : α → E) (a : α) (hfm : MeasureTheory.StronglyMeasurable f)
Goal: ∫ (x : α), f x ∂MeasureTheory.Measure.dirac a = f a
-- MeasureTheory.integral_dirac
@[simp]
theorem integral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α) :
    ∫ x, f x ∂Measure.dirac a = f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
Goal: ∫ (x : α), f x ∂MeasureTheory.Measure.dirac a = f a
-- MeasureTheory.setIntegral_dirac'
theorem setIntegral_dirac' {mα : MeasurableSpace α} {f : α → E} (hf : StronglyMeasurable f) (a : α)
    {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {mα : MeasurableSpace α} {f : α → E} (hf : MeasureTheory.StronglyMeasurable f) (a : α) {s : Set α} (hs : MeasurableSet s) [Decidable (a ∈ s)]
Goal: ∫ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0
-- MeasureTheory.setIntegral_dirac
theorem setIntegral_dirac [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α)
    (s : Set α) [Decidable (a ∈ s)] :
    ∫ x in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] [MeasurableSpace α] [MeasurableSingletonClass α] (f : α → E) (a : α) (s : Set α) [Decidable (a ∈ s)]
Goal: ∫ (x : α) in s, f x ∂MeasureTheory.Measure.dirac a = if a ∈ s then f a else 0
-- MeasureTheory.mul_meas_ge_le_integral_of_nonneg
theorem mul_meas_ge_le_integral_of_nonneg {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f)
    (hf_int : Integrable f μ) (ε : ℝ) : ε * (μ { x | ε ≤ f x }).toReal ≤ ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f) (hf_int : MeasureTheory.Integrable f μ) (ε : ℝ)
Goal: ε * (μ {x | ε ≤ f x}).toReal ≤ ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_mul_norm_le_Lp_mul_Lq
theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ}
    (hpq : p.IsConjExponent q) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ a, ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ a, ‖g a‖ ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {E : Type u_7} [NormedAddCommGroup E] {f g : α → E} {p q : ℝ} (hpq : p.IsConjExponent q) (hf : MeasureTheory.Memℒp f (ENNReal.ofReal p) μ) (hg : MeasureTheory.Memℒp g (ENNReal.ofReal q) μ)
Goal: ∫ (a : α), ‖f a‖ * ‖g a‖ ∂μ ≤ (∫ (a : α), ‖f a‖ ^ p ∂μ) ^ (1 / p) * (∫ (a : α), ‖g a‖ ^ q ∂μ) ^ (1 / q)
-- MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg
theorem integral_mul_le_Lp_mul_Lq_of_nonneg {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ}
    (hf_nonneg : 0 ≤ᵐ[μ] f) (hg_nonneg : 0 ≤ᵐ[μ] g) (hf : Memℒp f (ENNReal.ofReal p) μ)
    (hg : Memℒp g (ENNReal.ofReal q) μ) :
    ∫ a, f a * g a ∂μ ≤ (∫ a, f a ^ p ∂μ) ^ (1 / p) * (∫ a, g a ^ q ∂μ) ^ (1 / q) 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {p q : ℝ} (hpq : p.IsConjExponent q) {f g : α → ℝ} (hf_nonneg : 0 ≤ᵐ[μ] f) (hg_nonneg : 0 ≤ᵐ[μ] g) (hf : MeasureTheory.Memℒp f (ENNReal.ofReal p) μ) (hg : MeasureTheory.Memℒp g (ENNReal.ofReal q) μ)
Goal: ∫ (a : α), f a * g a ∂μ ≤ (∫ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫ (a : α), g a ^ q ∂μ) ^ (1 / q)
-- MeasureTheory.integral_countable'
theorem integral_countable' [Countable α] [MeasurableSingletonClass α] {μ : Measure α}
    {f : α → E} (hf : Integrable f μ) :
    ∫ a, f a ∂μ = ∑' a, (μ {a}).toReal • f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} [Countable α] [MeasurableSingletonClass α] {μ : MeasureTheory.Measure α} {f : α → E} (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (a : α), f a ∂μ = ∑' (a : α), (μ {a}).toReal • f a
-- MeasureTheory.integral_singleton'
theorem integral_singleton' {μ : Measure α} {f : α → E} (hf : StronglyMeasurable f) (a : α) :
    ∫ a in {a}, f a ∂μ = (μ {a}).toReal • f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → E} (hf : MeasureTheory.StronglyMeasurable f) (a : α)
Goal: ∫ (a : α) in {a}, f a ∂μ = (μ {a}).toReal • f a
-- MeasureTheory.integral_singleton
theorem integral_singleton [MeasurableSingletonClass α] {μ : Measure α} (f : α → E) (a : α) :
    ∫ a in {a}, f a ∂μ = (μ {a}).toReal • f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} [MeasurableSingletonClass α] {μ : MeasureTheory.Measure α} (f : α → E) (a : α)
Goal: ∫ (a : α) in {a}, f a ∂μ = (μ {a}).toReal • f a
-- MeasureTheory.integral_countable
theorem integral_countable [MeasurableSingletonClass α] (f : α → E) {s : Set α} (hs : s.Countable)
    (hf : Integrable f (μ.restrict s)) :
    ∫ a in s, f a ∂μ = ∑' a : s, (μ {(a : α)}).toReal • f a 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (f : α → E) {s : Set α} (hs : s.Countable) (hf : MeasureTheory.Integrable f (μ.restrict s))
Goal: ∫ (a : α) in s, f a ∂μ = ∑' (a : ↑s), (μ {↑a}).toReal • f ↑a
-- MeasureTheory.integral_finset
theorem integral_finset [MeasurableSingletonClass α] (s : Finset α) (f : α → E)
    (hf : Integrable f (μ.restrict s)) :
    ∫ x in s, f x ∂μ = ∑ x ∈ s, (μ {x}).toReal • f x 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] (s : Finset α) (f : α → E) (hf : MeasureTheory.Integrable f (μ.restrict ↑s))
Goal: ∫ (x : α) in ↑s, f x ∂μ = ∑ x ∈ s, (μ {x}).toReal • f x
-- MeasureTheory.integral_fintype
theorem integral_fintype [MeasurableSingletonClass α] [Fintype α] (f : α → E)
    (hf : Integrable f μ) :
    ∫ x, f x ∂μ = ∑ x, (μ {x}).toReal • f x 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] [Fintype α] (f : α → E) (hf : MeasureTheory.Integrable f μ)
Goal: ∫ (x : α), f x ∂μ = ∑ x : α, (μ {x}).toReal • f x
-- MeasureTheory.integral_unique
theorem integral_unique [Unique α] (f : α → E) : ∫ x, f x ∂μ = (μ univ).toReal • f default 
Arguments: {α : Type u_1} {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [hE : CompleteSpace E] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Unique α] (f : α → E)
Goal: ∫ (x : α), f x ∂μ = (μ Set.univ).toReal • f default
-- MeasureTheory.integral_pos_of_integrable_nonneg_nonzero
theorem integral_pos_of_integrable_nonneg_nonzero [TopologicalSpace α] [Measure.IsOpenPosMeasure μ]
    {f : α → ℝ} {x : α} (f_cont : Continuous f) (f_int : Integrable f μ) (f_nonneg : 0 ≤ f)
    (f_x : f x ≠ 0) : 0 < ∫ x, f x ∂μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace α] [μ.IsOpenPosMeasure] {f : α → ℝ} {x : α} (f_cont : Continuous f) (f_int : MeasureTheory.Integrable f μ) (f_nonneg : 0 ≤ f) (f_x : f x ≠ 0)
Goal: 0 < ∫ (x : α), f x ∂μ
-- MeasureTheory.SimpleFunc.coe_toLargerSpace_eq
theorem SimpleFunc.coe_toLargerSpace_eq (hm : m ≤ m0) (f : @SimpleFunc β m γ) :
    ⇑(f.toLargerSpace hm) = f 
Arguments: {β : Type u_7} {γ : Type u_8} {m m0 : MeasurableSpace β} (hm : m ≤ m0) (f : MeasureTheory.SimpleFunc β γ)
Goal: ↑(MeasureTheory.SimpleFunc.toLargerSpace hm f) = ↑f
-- MeasureTheory.integral_simpleFunc_larger_space
theorem integral_simpleFunc_larger_space (hm : m ≤ m0) (f : @SimpleFunc β m F)
    (hf_int : Integrable f μ) :
    ∫ x, f x ∂μ = ∑ x ∈ @SimpleFunc.range β F m f, ENNReal.toReal (μ (f ⁻¹' {x})) • x 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {β : Type u_7} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} (hm : m ≤ m0) (f : MeasureTheory.SimpleFunc β F) (hf_int : MeasureTheory.Integrable (↑f) μ)
Goal: ∫ (x : β), ↑f x ∂μ = ∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • x
-- MeasureTheory.integral_trim_simpleFunc
theorem integral_trim_simpleFunc (hm : m ≤ m0) (f : @SimpleFunc β m F) (hf_int : Integrable f μ) :
    ∫ x, f x ∂μ = ∫ x, f x ∂μ.trim hm 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F] {β : Type u_7} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} (hm : m ≤ m0) (f : MeasureTheory.SimpleFunc β F) (hf_int : MeasureTheory.Integrable (↑f) μ)
Goal: ∫ (x : β), ↑f x ∂μ = ∫ (x : β), ↑f x ∂μ.trim hm
-- MeasureTheory.integral_trim
theorem integral_trim (hm : m ≤ m0) {f : β → G} (hf : StronglyMeasurable[m] f) :
    ∫ x, f x ∂μ = ∫ x, f x ∂μ.trim hm 
Arguments: {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {β : Type u_7} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} (hm : m ≤ m0) {f : β → G} (hf : MeasureTheory.StronglyMeasurable f)
Goal: ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm
-- MeasureTheory.integral_trim_ae
theorem integral_trim_ae (hm : m ≤ m0) {f : β → G} (hf : AEStronglyMeasurable f (μ.trim hm)) :
    ∫ x, f x ∂μ = ∫ x, f x ∂μ.trim hm 
Arguments: {G : Type u_5} [NormedAddCommGroup G] [NormedSpace ℝ G] {β : Type u_7} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} (hm : m ≤ m0) {f : β → G} (hf : MeasureTheory.AEStronglyMeasurable f (μ.trim hm))
Goal: ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm
-- MeasureTheory.ae_eq_trim_of_stronglyMeasurable
theorem ae_eq_trim_of_stronglyMeasurable [TopologicalSpace γ] [MetrizableSpace γ] (hm : m ≤ m0)
    {f g : β → γ} (hf : StronglyMeasurable[m] f) (hg : StronglyMeasurable[m] g)
    (hfg : f =ᵐ[μ] g) : f =ᵐ[μ.trim hm] g 
Arguments: {β : Type u_7} {γ : Type u_8} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} [TopologicalSpace γ] [TopologicalSpace.MetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g) (hfg : f =ᵐ[μ] g)
Goal: f =ᵐ[μ.trim hm] g
-- MeasureTheory.ae_eq_trim_iff
theorem ae_eq_trim_iff [TopologicalSpace γ] [MetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ}
    (hf : StronglyMeasurable[m] f) (hg : StronglyMeasurable[m] g) :
    f =ᵐ[μ.trim hm] g ↔ f =ᵐ[μ] g 
Arguments: {β : Type u_7} {γ : Type u_8} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} [TopologicalSpace γ] [TopologicalSpace.MetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: f =ᵐ[μ.trim hm] g ↔ f =ᵐ[μ] g
-- MeasureTheory.ae_le_trim_of_stronglyMeasurable
theorem ae_le_trim_of_stronglyMeasurable [LinearOrder γ] [TopologicalSpace γ]
    [OrderClosedTopology γ] [PseudoMetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ}
    (hf : StronglyMeasurable[m] f) (hg : StronglyMeasurable[m] g) (hfg : f ≤ᵐ[μ] g) :
    f ≤ᵐ[μ.trim hm] g 
Arguments: {β : Type u_7} {γ : Type u_8} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} [LinearOrder γ] [TopologicalSpace γ] [OrderClosedTopology γ] [TopologicalSpace.PseudoMetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g) (hfg : f ≤ᵐ[μ] g)
Goal: f ≤ᵐ[μ.trim hm] g
-- MeasureTheory.ae_le_trim_iff
theorem ae_le_trim_iff [LinearOrder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
    [PseudoMetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ} (hf : StronglyMeasurable[m] f)
    (hg : StronglyMeasurable[m] g) : f ≤ᵐ[μ.trim hm] g ↔ f ≤ᵐ[μ] g 
Arguments: {β : Type u_7} {γ : Type u_8} {m m0 : MeasurableSpace β} {μ : MeasureTheory.Measure β} [LinearOrder γ] [TopologicalSpace γ] [OrderClosedTopology γ] [TopologicalSpace.PseudoMetrizableSpace γ] (hm : m ≤ m0) {f g : β → γ} (hf : MeasureTheory.StronglyMeasurable f) (hg : MeasureTheory.StronglyMeasurable g)
Goal: f ≤ᵐ[μ.trim hm] g ↔ f ≤ᵐ[μ] g
-- MeasureTheory.eLpNorm_one_le_of_le
theorem eLpNorm_one_le_of_le {r : ℝ≥0} (hfint : Integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ)
    (hf : ∀ᵐ ω ∂μ, f ω ≤ r) : eLpNorm f 1 μ ≤ 2 * μ Set.univ * r 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} {r : ℝ≥0} (hfint : MeasureTheory.Integrable f μ) (hfint' : 0 ≤ ∫ (x : α), f x ∂μ) (hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ ↑r)
Goal: MeasureTheory.eLpNorm f 1 μ ≤ 2 * μ Set.univ * ↑r
-- MeasureTheory.eLpNorm_one_le_of_le'
theorem eLpNorm_one_le_of_le' {r : ℝ} (hfint : Integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ)
    (hf : ∀ᵐ ω ∂μ, f ω ≤ r) : eLpNorm f 1 μ ≤ 2 * μ Set.univ * ENNReal.ofReal r 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ℝ} {r : ℝ} (hfint : MeasureTheory.Integrable f μ) (hfint' : 0 ≤ ∫ (x : α), f x ∂μ) (hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r)
Goal: MeasureTheory.eLpNorm f 1 μ ≤ 2 * μ Set.univ * ENNReal.ofReal r
-- BoxIntegral.Prepartition.mem_splitCenter
@[simp]
theorem mem_splitCenter : J ∈ splitCenter I ↔ ∃ s, I.splitCenterBox s = J 
Arguments: {ι : Type u_1} [Fintype ι] {I J : BoxIntegral.Box ι}
Goal: J ∈ BoxIntegral.Prepartition.splitCenter I ↔ ∃ s, I.splitCenterBox s = J
-- BoxIntegral.Prepartition.isPartition_splitCenter
theorem isPartition_splitCenter (I : Box ι) : IsPartition (splitCenter I) 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι)
Goal: (BoxIntegral.Prepartition.splitCenter I).IsPartition
-- BoxIntegral.Prepartition.upper_sub_lower_of_mem_splitCenter
theorem upper_sub_lower_of_mem_splitCenter (h : J ∈ splitCenter I) (i : ι) :
    J.upper i - J.lower i = (I.upper i - I.lower i) / 2 
Arguments: {ι : Type u_1} [Fintype ι] {I J : BoxIntegral.Box ι} (h : J ∈ BoxIntegral.Prepartition.splitCenter I) (i : ι)
Goal: J.upper i - J.lower i = (I.upper i - I.lower i) / 2
-- BoxIntegral.Box.subbox_induction_on
@[elab_as_elim]
theorem subbox_induction_on {p : Box ι → Prop} (I : Box ι)
    (H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J)
    (H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ),
      z ∈ Box.Icc J → Box.Icc J ⊆ U →
        (∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
    p I 
Arguments: {ι : Type u_1} [Fintype ι] {p : BoxIntegral.Box ι → Prop} (I : BoxIntegral.Box ι) (H_ind : ∀ J ≤ I, (∀ J' ∈ BoxIntegral.Prepartition.splitCenter J, p J') → p J) (H_nhds :
  ∀ z ∈ BoxIntegral.Box.Icc I,
    ∃ U ∈ nhdsWithin z (BoxIntegral.Box.Icc I),
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ BoxIntegral.Box.Icc J →
            BoxIntegral.Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J)
Goal: p I
-- BoxIntegral.Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic
theorem exists_taggedPartition_isHenstock_isSubordinate_homothetic (I : Box ι)
    (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    ∃ π : TaggedPrepartition I, π.IsPartition ∧ π.IsHenstock ∧ π.IsSubordinate r ∧
      (∀ J ∈ π, ∃ m : ℕ, ∀ i, (J : _).upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) ∧
        π.distortion = I.distortion 
Arguments: {ι : Type u_1} [Fintype ι] (I : BoxIntegral.Box ι) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: ∃ π,
  π.IsPartition ∧
    π.IsHenstock ∧
      π.IsSubordinate r ∧
        (∀ J ∈ π, ∃ m, ∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) ∧ π.distortion = I.distortion
-- BoxIntegral.Prepartition.exists_tagged_le_isHenstock_isSubordinate_iUnion_eq
theorem exists_tagged_le_isHenstock_isSubordinate_iUnion_eq {I : Box ι} (r : (ι → ℝ) → Ioi (0 : ℝ))
    (π : Prepartition I) :
    ∃ π' : TaggedPrepartition I, π'.toPrepartition ≤ π ∧ π'.IsHenstock ∧ π'.IsSubordinate r ∧
      π'.distortion = π.distortion ∧ π'.iUnion = π.iUnion 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (r : (ι → ℝ) → ↑(Set.Ioi 0)) (π : BoxIntegral.Prepartition I)
Goal: ∃ π', π'.toPrepartition ≤ π ∧ π'.IsHenstock ∧ π'.IsSubordinate r ∧ π'.distortion = π.distortion ∧ π'.iUnion = π.iUnion
-- BoxIntegral.Prepartition.toSubordinate_toPrepartition_le
theorem toSubordinate_toPrepartition_le (π : Prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π.toSubordinate r).toPrepartition ≤ π 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π.toSubordinate r).toPrepartition ≤ π
-- BoxIntegral.Prepartition.isHenstock_toSubordinate
theorem isHenstock_toSubordinate (π : Prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π.toSubordinate r).IsHenstock 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π.toSubordinate r).IsHenstock
-- BoxIntegral.Prepartition.isSubordinate_toSubordinate
theorem isSubordinate_toSubordinate (π : Prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π.toSubordinate r).IsSubordinate r 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π.toSubordinate r).IsSubordinate r
-- BoxIntegral.Prepartition.distortion_toSubordinate
@[simp]
theorem distortion_toSubordinate (π : Prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π.toSubordinate r).distortion = π.distortion 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π.toSubordinate r).distortion = π.distortion
-- BoxIntegral.Prepartition.iUnion_toSubordinate
@[simp]
theorem iUnion_toSubordinate (π : Prepartition I) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π.toSubordinate r).iUnion = π.iUnion 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π : BoxIntegral.Prepartition I) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π.toSubordinate r).iUnion = π.iUnion
-- BoxIntegral.TaggedPrepartition.isPartition_unionComplToSubordinate
theorem isPartition_unionComplToSubordinate (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
    (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    IsPartition (π₁.unionComplToSubordinate π₂ hU r) 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π₁ : BoxIntegral.TaggedPrepartition I) (π₂ : BoxIntegral.Prepartition I) (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π₁.unionComplToSubordinate π₂ hU r).IsPartition
-- BoxIntegral.TaggedPrepartition.unionComplToSubordinate_boxes
@[simp]
theorem unionComplToSubordinate_boxes (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
    (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π₁.unionComplToSubordinate π₂ hU r).boxes = π₁.boxes ∪ (π₂.toSubordinate r).boxes 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π₁ : BoxIntegral.TaggedPrepartition I) (π₂ : BoxIntegral.Prepartition I) (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π₁.unionComplToSubordinate π₂ hU r).boxes = π₁.boxes ∪ (π₂.toSubordinate r).boxes
-- BoxIntegral.TaggedPrepartition.iUnion_unionComplToSubordinate_boxes
@[simp]
theorem iUnion_unionComplToSubordinate_boxes (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
    (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π₁.unionComplToSubordinate π₂ hU r).iUnion = I 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π₁ : BoxIntegral.TaggedPrepartition I) (π₂ : BoxIntegral.Prepartition I) (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π₁.unionComplToSubordinate π₂ hU r).iUnion = ↑I
-- BoxIntegral.TaggedPrepartition.distortion_unionComplToSubordinate
@[simp]
theorem distortion_unionComplToSubordinate (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
    (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
    (π₁.unionComplToSubordinate π₂ hU r).distortion = max π₁.distortion π₂.distortion 
Arguments: {ι : Type u_1} [Fintype ι] {I : BoxIntegral.Box ι} (π₁ : BoxIntegral.TaggedPrepartition I) (π₂ : BoxIntegral.Prepartition I) (hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → ↑(Set.Ioi 0))
Goal: (π₁.unionComplToSubordinate π₂ hU r).distortion = max π₁.distortion π₂.distortion
-- BoxIntegral.TaggedPrepartition.tag_mem_Icc
structure TaggedPrepartition (I : Box ι) extends Prepartition I where
    tag : Box ι → ι → ℝ
    tag_mem_Icc : ∀ J, tag J ∈ Box.Icc I
Arguments: {ι : Type u_1} {I : BoxIntegral.Box ι} (self : BoxIntegral.TaggedPrepartition I) (J : BoxIntegral.Box ι)
Goal: self.tag J ∈ BoxIntegral.Box.Icc I
-- BoxIntegral.TaggedPrepartition.mem_toPrepartition
@[simp]
theorem mem_toPrepartition {π : TaggedPrepartition I} : J ∈ π.toPrepartition ↔ J ∈ π 
Arguments: {ι : Type u_1} {I J : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I}
Goal: J ∈ π.toPrepartition ↔ J ∈ π