ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Measurable.forall
lemma Measurable.forall [Countable ι] {p : ι → α → Prop} (hp : ∀ i, Measurable (p i)) :
    Measurable fun a ↦ ∀ i, p i a 
Arguments: {α : Type u_1} {ι : Sort uι} [MeasurableSpace α] [Countable ι] {p : ι → α → Prop} (hp : ∀ (i : ι), Measurable (p i))
Goal: Measurable fun a => ∀ (i : ι), p i a
-- Measurable.exists
lemma Measurable.exists [Countable ι] {p : ι → α → Prop} (hp : ∀ i, Measurable (p i)) :
    Measurable fun a ↦ ∃ i, p i a 
Arguments: {α : Type u_1} {ι : Sort uι} [MeasurableSpace α] [Countable ι] {p : ι → α → Prop} (hp : ∀ (i : ι), Measurable (p i))
Goal: Measurable fun a => ∃ i, p i a
-- measurable_set_iff
lemma measurable_set_iff : Measurable g ↔ ∀ a, Measurable fun x ↦ a ∈ g x 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace β] {g : β → Set α}
Goal: Measurable g ↔ ∀ (a : α), Measurable fun x => a ∈ g x
-- measurable_set_mem
@[aesop safe 100 apply (rule_sets := [Measurable])]
lemma measurable_set_mem (a : α) : Measurable fun s : Set α ↦ a ∈ s 
Arguments: {α : Type u_1} (a : α)
Goal: Measurable fun s => a ∈ s
-- measurable_set_not_mem
@[aesop safe 100 apply (rule_sets := [Measurable])]
lemma measurable_set_not_mem (a : α) : Measurable fun s : Set α ↦ a ∉ s 
Arguments: {α : Type u_1} (a : α)
Goal: Measurable fun s => a ∉ s
-- measurableSet_mem
@[aesop safe 100 apply (rule_sets := [Measurable])]
lemma measurableSet_mem (a : α) : MeasurableSet {s : Set α | a ∈ s} 
Arguments: {α : Type u_1} (a : α)
Goal: MeasurableSet {s | a ∈ s}
-- measurableSet_not_mem
@[aesop safe 100 apply (rule_sets := [Measurable])]
lemma measurableSet_not_mem (a : α) : MeasurableSet {s : Set α | a ∉ s} 
Arguments: {α : Type u_1} (a : α)
Goal: MeasurableSet {s | a ∉ s}
-- measurable_compl
lemma measurable_compl : Measurable ((·ᶜ) : Set α → Set α) 
Arguments: {α : Type u_1}
Goal: Measurable fun x => xᶜ
-- MeasurableSpace.generateFrom_singleton
@[simp] theorem generateFrom_singleton (s : Set α) :
    generateFrom {s} = MeasurableSpace.comap (· ∈ s) ⊤ 
Arguments: {α : Type u_1} (s : Set α)
Goal: MeasurableSpace.generateFrom {s} = MeasurableSpace.comap (fun x => x ∈ s) ⊤
-- Filter.IsMeasurablyGenerated.exists_measurable_subset
class IsMeasurablyGenerated (f : Filter α) : Prop where
  exists_measurable_subset : ∀ ⦃s⦄, s ∈ f → ∃ t ∈ f, MeasurableSet t ∧ t ⊆ s
Arguments: {α : Type u_1} [MeasurableSpace α] {f : Filter α} [self : f.IsMeasurablyGenerated] ⦃s : Set α⦄
Goal: s ∈ f → ∃ t ∈ f, MeasurableSet t ∧ t ⊆ s
-- Filter.Eventually.exists_measurable_mem
theorem Eventually.exists_measurable_mem {f : Filter α} [IsMeasurablyGenerated f] {p : α → Prop}
    (h : ∀ᶠ x in f, p x) : ∃ s ∈ f, MeasurableSet s ∧ ∀ x ∈ s, p x 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : Filter α} [f.IsMeasurablyGenerated] {p : α → Prop} (h : ∀ᶠ (x : α) in f, p x)
Goal: ∃ s ∈ f, MeasurableSet s ∧ ∀ x ∈ s, p x
-- Filter.Eventually.exists_measurable_mem_of_smallSets
theorem Eventually.exists_measurable_mem_of_smallSets {f : Filter α} [IsMeasurablyGenerated f]
    {p : Set α → Prop} (h : ∀ᶠ s in f.smallSets, p s) : ∃ s ∈ f, MeasurableSet s ∧ p s 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : Filter α} [f.IsMeasurablyGenerated] {p : Set α → Prop} (h : ∀ᶠ (s : Set α) in f.smallSets, p s)
Goal: ∃ s ∈ f, MeasurableSet s ∧ p s
-- Filter.principal_isMeasurablyGenerated_iff
theorem principal_isMeasurablyGenerated_iff {s : Set α} :
    IsMeasurablyGenerated (𝓟 s) ↔ MeasurableSet s 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : Set α}
Goal: (𝓟 s).IsMeasurablyGenerated ↔ MeasurableSet s
-- measurableSet_tendsto
@[measurability]
lemma measurableSet_tendsto {_ : MeasurableSpace β} [MeasurableSpace γ]
    [Countable δ] {l : Filter δ} [l.IsCountablyGenerated]
    (l' : Filter γ) [l'.IsCountablyGenerated] [hl' : l'.IsMeasurablyGenerated]
    {f : δ → β → γ} (hf : ∀ i, Measurable (f i)) :
    MeasurableSet { x | Tendsto (fun n ↦ f n x) l l' } 
Arguments: {β : Type u_2} {γ : Type u_3} {δ : Type u_4}
Goal: ∀ {x : MeasurableSpace β} [inst : MeasurableSpace γ] [inst_1 : Countable δ] {l : Filter δ}
  [inst_2 : l.IsCountablyGenerated] (l' : Filter γ) [inst_3 : l'.IsCountablyGenerated] [hl' : l'.IsMeasurablyGenerated]
  {f : δ → β → γ}, (∀ (i : δ), Measurable (f i)) → MeasurableSet {x | Filter.Tendsto (fun n => f n x) l l'}
-- isCountablySpanning_measurableSet
theorem isCountablySpanning_measurableSet [MeasurableSpace α] :
    IsCountablySpanning { s : Set α | MeasurableSet s } 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: IsCountablySpanning {s | MeasurableSet s}
-- MeasurableSet.mem_coe
@[simp]
theorem mem_coe (a : α) (s : Subtype (MeasurableSet : Set α → Prop)) : a ∈ (s : Set α) ↔ a ∈ s 
Arguments: {α : Type u_1} [MeasurableSpace α] (a : α) (s : Subtype MeasurableSet)
Goal: a ∈ ↑s ↔ a ∈ s
-- MeasurableSet.coe_empty
@[simp]
theorem coe_empty : ↑(∅ : Subtype (MeasurableSet : Set α → Prop)) = (∅ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ↑∅ = ∅
-- MeasurableSet.coe_insert
@[simp]
theorem coe_insert [MeasurableSingletonClass α] (a : α)
    (s : Subtype (MeasurableSet : Set α → Prop)) :
    ↑(Insert.insert a s) = (Insert.insert a s : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α) (s : Subtype MeasurableSet)
Goal: ↑(insert a s) = insert a ↑s
-- MeasurableSet.coe_singleton
@[simp] theorem coe_singleton [MeasurableSingletonClass α] (a : α) :
    ↑({a} : Subtype (MeasurableSet : Set α → Prop)) = ({a} : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (a : α)
Goal: ↑{a} = {a}
-- MeasurableSet.coe_compl
@[simp]
theorem coe_compl (s : Subtype (MeasurableSet : Set α → Prop)) : ↑sᶜ = (sᶜ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] (s : Subtype MeasurableSet)
Goal: ↑sᶜ = (↑s)ᶜ
-- MeasurableSet.coe_union
@[simp]
theorem coe_union (s t : Subtype (MeasurableSet : Set α → Prop)) : ↑(s ∪ t) = (s ∪ t : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : Subtype MeasurableSet)
Goal: ↑(s ∪ t) = ↑s ∪ ↑t
-- MeasurableSet.sup_eq_union
@[simp]
protected theorem sup_eq_union (s t : {s : Set α // MeasurableSet s}) : s ⊔ t = s ∪ t 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : { s // MeasurableSet s })
Goal: s ⊔ t = s ∪ t
-- MeasurableSet.coe_inter
@[simp]
theorem coe_inter (s t : Subtype (MeasurableSet : Set α → Prop)) : ↑(s ∩ t) = (s ∩ t : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : Subtype MeasurableSet)
Goal: ↑(s ∩ t) = ↑s ∩ ↑t
-- MeasurableSet.inf_eq_inter
@[simp]
protected theorem inf_eq_inter (s t : {s : Set α // MeasurableSet s}) : s ⊓ t = s ∩ t 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : { s // MeasurableSet s })
Goal: s ⊓ t = s ∩ t
-- MeasurableSet.coe_sdiff
@[simp]
theorem coe_sdiff (s t : Subtype (MeasurableSet : Set α → Prop)) : ↑(s \ t) = (s : Set α) \ t 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : Subtype MeasurableSet)
Goal: ↑(s \ t) = ↑s \ ↑t
-- MeasurableSet.coe_himp
@[simp]
lemma coe_himp (s t : Subtype (MeasurableSet : Set α → Prop)) : ↑(s ⇨ t) = (s ⇨ t : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α] (s t : Subtype MeasurableSet)
Goal: ↑(s ⇨ t) = ↑s ⇨ ↑t
-- MeasurableSet.coe_bot
@[simp]
theorem coe_bot : ↑(⊥ : Subtype (MeasurableSet : Set α → Prop)) = (⊥ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ↑⊥ = ⊥
-- MeasurableSet.coe_top
@[simp]
theorem coe_top : ↑(⊤ : Subtype (MeasurableSet : Set α → Prop)) = (⊤ : Set α) 
Arguments: {α : Type u_1} [MeasurableSpace α]
Goal: ↑⊤ = ⊤
-- MeasurableSet.measurableSet_blimsup
@[measurability]
theorem measurableSet_blimsup {s : ℕ → Set α} {p : ℕ → Prop} (h : ∀ n, p n → MeasurableSet (s n)) :
    MeasurableSet <| blimsup s atTop p 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : ℕ → Set α} {p : ℕ → Prop} (h : ∀ (n : ℕ), p n → MeasurableSet (s n))
Goal: MeasurableSet (Filter.blimsup s Filter.atTop p)
-- MeasurableSet.measurableSet_bliminf
@[measurability]
theorem measurableSet_bliminf {s : ℕ → Set α} {p : ℕ → Prop} (h : ∀ n, p n → MeasurableSet (s n)) :
    MeasurableSet <| Filter.bliminf s Filter.atTop p 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : ℕ → Set α} {p : ℕ → Prop} (h : ∀ (n : ℕ), p n → MeasurableSet (s n))
Goal: MeasurableSet (Filter.bliminf s Filter.atTop p)
-- MeasurableSet.measurableSet_limsup
@[measurability]
theorem measurableSet_limsup {s : ℕ → Set α} (hs : ∀ n, MeasurableSet <| s n) :
    MeasurableSet <| Filter.limsup s Filter.atTop 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : ℕ → Set α} (hs : ∀ (n : ℕ), MeasurableSet (s n))
Goal: MeasurableSet (Filter.limsup s Filter.atTop)
-- MeasurableSet.measurableSet_liminf
@[measurability]
theorem measurableSet_liminf {s : ℕ → Set α} (hs : ∀ n, MeasurableSet <| s n) :
    MeasurableSet <| Filter.liminf s Filter.atTop 
Arguments: {α : Type u_1} [MeasurableSpace α] {s : ℕ → Set α} (hs : ∀ (n : ℕ), MeasurableSet (s n))
Goal: MeasurableSet (Filter.liminf s Filter.atTop)
-- MeasureTheory.Measure.integral_comp_smul
theorem integral_comp_smul (f : E → F) (R : ℝ) :
    ∫ x, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) (R : ℝ)
Goal: ∫ (x : E), f (R • x) ∂μ = |(R ^ FiniteDimensional.finrank ℝ E)⁻¹| • ∫ (x : E), f x ∂μ
-- MeasureTheory.Measure.integral_comp_smul_of_nonneg
theorem integral_comp_smul_of_nonneg (f : E → F) (R : ℝ) {hR : 0 ≤ R} :
    ∫ x, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) (R : ℝ) {hR : 0 ≤ R}
Goal: ∫ (x : E), f (R • x) ∂μ = (R ^ FiniteDimensional.finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ
-- MeasureTheory.Measure.integral_comp_inv_smul
theorem integral_comp_inv_smul (f : E → F) (R : ℝ) :
    ∫ x, f (R⁻¹ • x) ∂μ = |R ^ finrank ℝ E| • ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) (R : ℝ)
Goal: ∫ (x : E), f (R⁻¹ • x) ∂μ = |R ^ FiniteDimensional.finrank ℝ E| • ∫ (x : E), f x ∂μ
-- MeasureTheory.Measure.integral_comp_inv_smul_of_nonneg
theorem integral_comp_inv_smul_of_nonneg (f : E → F) {R : ℝ} (hR : 0 ≤ R) :
    ∫ x, f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ x, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) {R : ℝ} (hR : 0 ≤ R)
Goal: ∫ (x : E), f (R⁻¹ • x) ∂μ = R ^ FiniteDimensional.finrank ℝ E • ∫ (x : E), f x ∂μ
-- MeasureTheory.Measure.setIntegral_comp_smul
theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
    ∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0)
Goal: ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ FiniteDimensional.finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ
-- MeasureTheory.Measure.setIntegral_comp_smul_of_pos
theorem setIntegral_comp_smul_of_pos (f : E → F) {R : ℝ} (s : Set E) (hR : 0 < R) :
    ∫ x in s, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x in R • s, f x ∂μ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E → F) {R : ℝ} (s : Set E) (hR : 0 < R)
Goal: ∫ (x : E) in s, f (R • x) ∂μ = (R ^ FiniteDimensional.finrank ℝ E)⁻¹ • ∫ (x : E) in R • s, f x ∂μ
-- MeasureTheory.Measure.integral_comp_mul_left
theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (a * x)) = |a⁻¹| • ∫ y : ℝ, g y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (g : ℝ → F) (a : ℝ)
Goal: ∫ (x : ℝ), g (a * x) = |a⁻¹| • ∫ (y : ℝ), g y
-- MeasureTheory.Measure.integral_comp_inv_mul_left
theorem integral_comp_inv_mul_left (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (a⁻¹ * x)) = |a| • ∫ y : ℝ, g y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (g : ℝ → F) (a : ℝ)
Goal: ∫ (x : ℝ), g (a⁻¹ * x) = |a| • ∫ (y : ℝ), g y
-- MeasureTheory.Measure.integral_comp_mul_right
theorem integral_comp_mul_right (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (x * a)) = |a⁻¹| • ∫ y : ℝ, g y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (g : ℝ → F) (a : ℝ)
Goal: ∫ (x : ℝ), g (x * a) = |a⁻¹| • ∫ (y : ℝ), g y
-- MeasureTheory.Measure.integral_comp_inv_mul_right
theorem integral_comp_inv_mul_right (g : ℝ → F) (a : ℝ) :
    (∫ x : ℝ, g (x * a⁻¹)) = |a| • ∫ y : ℝ, g y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (g : ℝ → F) (a : ℝ)
Goal: ∫ (x : ℝ), g (x * a⁻¹) = |a| • ∫ (y : ℝ), g y
-- MeasureTheory.Measure.integral_comp_div
theorem integral_comp_div (g : ℝ → F) (a : ℝ) : (∫ x : ℝ, g (x / a)) = |a| • ∫ y : ℝ, g y 
Arguments: {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] (g : ℝ → F) (a : ℝ)
Goal: ∫ (x : ℝ), g (x / a) = |a| • ∫ (y : ℝ), g y
-- MeasureTheory.integrable_comp_smul_iff
theorem integrable_comp_smul_iff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
    (f : E → F) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ ↔ Integrable f μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] (μ : MeasureTheory.Measure E) [μ.IsAddHaarMeasure] (f : E → F) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => f (R • x)) μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.Integrable.comp_smul
theorem Integrable.comp_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
    [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : Measure E} [IsAddHaarMeasure μ]
    {f : E → F} (hf : Integrable f μ) {R : ℝ} (hR : R ≠ 0) : Integrable (fun x => f (R • x)) μ 
Arguments: {F : Type u_1} [NormedAddCommGroup F] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E] {μ : MeasureTheory.Measure E} [μ.IsAddHaarMeasure] {f : E → F} (hf : MeasureTheory.Integrable f μ) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => f (R • x)) μ
-- MeasureTheory.integrable_comp_mul_left_iff
theorem integrable_comp_mul_left_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (R * x)) ↔ Integrable g 
Arguments: {F : Type u_1} [NormedAddCommGroup F] (g : ℝ → F) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (R * x)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume
-- MeasureTheory.Integrable.comp_mul_left'
theorem Integrable.comp_mul_left' {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (R * x) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] {g : ℝ → F} (hg : MeasureTheory.Integrable g MeasureTheory.volume) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (R * x)) MeasureTheory.volume
-- MeasureTheory.integrable_comp_mul_right_iff
theorem integrable_comp_mul_right_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (x * R)) ↔ Integrable g 
Arguments: {F : Type u_1} [NormedAddCommGroup F] (g : ℝ → F) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (x * R)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume
-- MeasureTheory.Integrable.comp_mul_right'
theorem Integrable.comp_mul_right' {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (x * R) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] {g : ℝ → F} (hg : MeasureTheory.Integrable g MeasureTheory.volume) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (x * R)) MeasureTheory.volume
-- MeasureTheory.integrable_comp_div_iff
theorem integrable_comp_div_iff (g : ℝ → F) {R : ℝ} (hR : R ≠ 0) :
    (Integrable fun x => g (x / R)) ↔ Integrable g 
Arguments: {F : Type u_1} [NormedAddCommGroup F] (g : ℝ → F) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (x / R)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume
-- MeasureTheory.Integrable.comp_div
theorem Integrable.comp_div {g : ℝ → F} (hg : Integrable g) {R : ℝ} (hR : R ≠ 0) :
    Integrable fun x => g (x / R) 
Arguments: {F : Type u_1} [NormedAddCommGroup F] {g : ℝ → F} (hg : MeasureTheory.Integrable g MeasureTheory.volume) {R : ℝ} (hR : R ≠ 0)
Goal: MeasureTheory.Integrable (fun x => g (x / R)) MeasureTheory.volume
-- MeasureTheory.integrable_comp
theorem integrable_comp (g : F' → A) : Integrable (g ∘ f) ↔ Integrable g 
Arguments: {E' : Type u_2} {F' : Type u_3} {A : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace ℝ E'] [FiniteDimensional ℝ E'] [MeasurableSpace E'] [BorelSpace E'] [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] [FiniteDimensional ℝ F'] [MeasurableSpace F'] [BorelSpace F'] (f : E' ≃ₗᵢ[ℝ] F') [NormedAddCommGroup A] (g : F' → A)
Goal: MeasureTheory.Integrable (g ∘ ⇑f) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume
-- MeasureTheory.integral_comp
theorem integral_comp (g : F' → A) : ∫ (x : E'), g (f x) = ∫ (y : F'), g y 
Arguments: {E' : Type u_2} {F' : Type u_3} {A : Type u_4} [NormedAddCommGroup E'] [InnerProductSpace ℝ E'] [FiniteDimensional ℝ E'] [MeasurableSpace E'] [BorelSpace E'] [NormedAddCommGroup F'] [InnerProductSpace ℝ F'] [FiniteDimensional ℝ F'] [MeasurableSpace F'] [BorelSpace F'] (f : E' ≃ₗᵢ[ℝ] F') [NormedAddCommGroup A] [NormedSpace ℝ A] (g : F' → A)
Goal: ∫ (x : E'), g (f x) = ∫ (y : F'), g y
-- Orientation.measure_orthonormalBasis
theorem Orientation.measure_orthonormalBasis (o : Orientation ℝ F (Fin n))
    (b : OrthonormalBasis ι ℝ F) : o.volumeForm.measure (parallelepiped b) = 1 
Arguments: {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ F = n)] (o : Orientation ℝ F (Fin n)) (b : OrthonormalBasis ι ℝ F)
Goal: o.volumeForm.measure (parallelepiped ⇑b) = 1
-- Orientation.measure_eq_volume
theorem Orientation.measure_eq_volume (o : Orientation ℝ F (Fin n)) :
    o.volumeForm.measure = volume 
Arguments: {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {n : ℕ} [_i : Fact (FiniteDimensional.finrank ℝ F = n)] (o : Orientation ℝ F (Fin n))
Goal: o.volumeForm.measure = MeasureTheory.volume
-- OrthonormalBasis.volume_parallelepiped
theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
    volume (parallelepiped b) = 1 
Arguments: {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F)
Goal: MeasureTheory.volume (parallelepiped ⇑b) = 1
-- OrthonormalBasis.addHaar_eq_volume
theorem OrthonormalBasis.addHaar_eq_volume {ι F : Type*} [Fintype ι] [NormedAddCommGroup F]
    [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F]
    (b : OrthonormalBasis ι ℝ F) :
    b.toBasis.addHaar = volume 
Arguments: {ι : Type u_4} {F : Type u_5} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F)
Goal: b.toBasis.addHaar = MeasureTheory.volume
-- OrthonormalBasis.measurePreserving_measurableEquiv
theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.measurableEquiv volume volume 
Arguments: {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F)
Goal: MeasureTheory.MeasurePreserving (⇑b.measurableEquiv) MeasureTheory.volume MeasureTheory.volume
-- OrthonormalBasis.measurePreserving_repr
theorem OrthonormalBasis.measurePreserving_repr (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.repr volume volume 
Arguments: {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F)
Goal: MeasureTheory.MeasurePreserving (⇑b.repr) MeasureTheory.volume MeasureTheory.volume
-- OrthonormalBasis.measurePreserving_repr_symm
theorem OrthonormalBasis.measurePreserving_repr_symm (b : OrthonormalBasis ι ℝ F) :
    MeasurePreserving b.repr.symm volume volume 
Arguments: {ι : Type u_1} {F : Type u_3} [Fintype ι] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (b : OrthonormalBasis ι ℝ F)
Goal: MeasureTheory.MeasurePreserving (⇑b.repr.symm) MeasureTheory.volume MeasureTheory.volume
-- EuclideanSpace.volume_preserving_measurableEquiv
theorem EuclideanSpace.volume_preserving_measurableEquiv :
    MeasurePreserving (EuclideanSpace.measurableEquiv ι) 
Arguments: (ι : Type u_4) [Fintype ι]
Goal: MeasureTheory.MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι)) MeasureTheory.volume MeasureTheory.volume
-- PiLp.volume_preserving_equiv
theorem PiLp.volume_preserving_equiv : MeasurePreserving (WithLp.equiv 2 (ι → ℝ)) 
Arguments: (ι : Type u_4) [Fintype ι]
Goal: MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι → ℝ))) MeasureTheory.volume MeasureTheory.volume
-- PiLp.volume_preserving_equiv_symm
theorem PiLp.volume_preserving_equiv_symm : MeasurePreserving (WithLp.equiv 2 (ι → ℝ)).symm 
Arguments: (ι : Type u_4) [Fintype ι]
Goal: MeasureTheory.MeasurePreserving (⇑(WithLp.equiv 2 (ι → ℝ)).symm) MeasureTheory.volume MeasureTheory.volume
-- volume_euclideanSpace_eq_dirac
lemma volume_euclideanSpace_eq_dirac [IsEmpty ι] :
    (volume : Measure (EuclideanSpace ℝ ι)) = Measure.dirac 0 
Arguments: (ι : Type u_4) [Fintype ι] [IsEmpty ι]
Goal: MeasureTheory.volume = MeasureTheory.Measure.dirac 0
-- LinearIsometryEquiv.measurePreserving
theorem measurePreserving : MeasurePreserving f 
Arguments: {E : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (f : E ≃ₗᵢ[ℝ] F)
Goal: MeasureTheory.MeasurePreserving (⇑f) MeasureTheory.volume MeasureTheory.volume
-- LinearIsometryEquiv.coe_toMeasureEquiv
@[simp] theorem coe_toMeasureEquiv : (f.toMeasureEquiv : E → F) = f 
Arguments: {E : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [MeasurableSpace F] [BorelSpace F] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (f : E ≃ₗᵢ[ℝ] F)
Goal: ⇑f.toMeasureEquiv = ⇑f
-- LinearIsometryEquiv.toMeasureEquiv_symm
theorem toMeasureEquiv_symm : f.toMeasureEquiv.symm = f.symm.toMeasureEquiv 
Arguments: {E : Type u_2} {F : Type u_3} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [MeasurableSpace F] [BorelSpace F] [NormedAddCommGroup E] [InnerProductSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (f : E ≃ₗᵢ[ℝ] F)
Goal: f.toMeasureEquiv.symm = f.symm.toMeasureEquiv
-- MeasureTheory.llr_def
lemma llr_def (μ ν : Measure α) : llr μ ν = fun x ↦ log (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α)
Goal: MeasureTheory.llr μ ν = fun x => Real.log (μ.rnDeriv ν x).toReal
-- MeasureTheory.exp_llr
lemma exp_llr (μ ν : Measure α) [SigmaFinite μ] :
    (fun x ↦ exp (llr μ ν x))
      =ᵐ[ν] fun x ↦ if μ.rnDeriv ν x = 0 then 1 else (μ.rnDeriv ν x).toReal 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: (fun x => rexp (MeasureTheory.llr μ ν x)) =ᵐ[ν] fun x => if μ.rnDeriv ν x = 0 then 1 else (μ.rnDeriv ν x).toReal