ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.AEFinStronglyMeasurable.inf
@[aesop safe 20 (rule_sets := [Measurable])]
protected theorem inf [SemilatticeInf β] [ContinuousInf β] (hf : AEFinStronglyMeasurable f μ)
    (hg : AEFinStronglyMeasurable g μ) : AEFinStronglyMeasurable (f ⊓ g) μ 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f g : α → β} [Zero β] [SemilatticeInf β] [ContinuousInf β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ) (hg : MeasureTheory.AEFinStronglyMeasurable g μ)
Goal: MeasureTheory.AEFinStronglyMeasurable (f ⊓ g) μ
-- MeasureTheory.AEFinStronglyMeasurable.exists_set_sigmaFinite
theorem exists_set_sigmaFinite (hf : AEFinStronglyMeasurable f μ) :
    ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict tᶜ] 0 ∧ SigmaFinite (μ.restrict t) 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] [T2Space β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: ∃ t, MeasurableSet t ∧ f =ᵐ[μ.restrict tᶜ] 0 ∧ MeasureTheory.SigmaFinite (μ.restrict t)
-- MeasureTheory.AEFinStronglyMeasurable.measurableSet
protected theorem measurableSet (hf : AEFinStronglyMeasurable f μ) :
    MeasurableSet hf.sigmaFiniteSet 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] [T2Space β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: MeasurableSet hf.sigmaFiniteSet
-- MeasureTheory.AEFinStronglyMeasurable.ae_eq_zero_compl
theorem ae_eq_zero_compl (hf : AEFinStronglyMeasurable f μ) :
    f =ᵐ[μ.restrict hf.sigmaFiniteSetᶜ] 0 
Arguments: {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [TopologicalSpace β] {f : α → β} [Zero β] [T2Space β] (hf : MeasureTheory.AEFinStronglyMeasurable f μ)
Goal: f =ᵐ[μ.restrict hf.sigmaFiniteSetᶜ] 0
-- MeasureTheory.finStronglyMeasurable_iff_measurable
theorem finStronglyMeasurable_iff_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] : FinStronglyMeasurable f μ ↔ Measurable f 
Arguments: {α : Type u_1} {G : Type u_5} [SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {f : α → G} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.FinStronglyMeasurable f μ ↔ Measurable f
-- MeasureTheory.finStronglyMeasurable_of_measurable
@[aesop 90% apply (rule_sets := [Measurable])]
theorem finStronglyMeasurable_of_measurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] (hf : Measurable f) : FinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_5} [SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {f : α → G} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (hf : Measurable f)
Goal: MeasureTheory.FinStronglyMeasurable f μ
-- MeasureTheory.aefinStronglyMeasurable_iff_aemeasurable
theorem aefinStronglyMeasurable_iff_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] : AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_5} [SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {f : α → G} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: MeasureTheory.AEFinStronglyMeasurable f μ ↔ AEMeasurable f μ
-- MeasureTheory.aefinStronglyMeasurable_of_aemeasurable
@[aesop 90% apply (rule_sets := [Measurable])]
theorem aefinStronglyMeasurable_of_aemeasurable {_m0 : MeasurableSpace α} (μ : Measure α)
    [SigmaFinite μ] (hf : AEMeasurable f μ) : AEFinStronglyMeasurable f μ 
Arguments: {α : Type u_1} {G : Type u_5} [SeminormedAddCommGroup G] [MeasurableSpace G] [BorelSpace G] [SecondCountableTopology G] {f : α → G} {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] (hf : AEMeasurable f μ)
Goal: MeasureTheory.AEFinStronglyMeasurable f μ
-- MeasureTheory.measurable_uncurry_of_continuous_of_measurable
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type*} [TopologicalSpace ι]
    [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
    {mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
    {m : MeasurableSpace α} {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x)
    (h : ∀ i, Measurable (u i)) : Measurable (Function.uncurry u) 
Arguments: {α : Type u_5} {β : Type u_6} {ι : Type u_7} [TopologicalSpace ι] [TopologicalSpace.MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι] {mβ : MeasurableSpace β} [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [BorelSpace β] {m : MeasurableSpace α} {u : ι → α → β} (hu_cont : ∀ (x : α), Continuous fun i => u i x) (h : ∀ (i : ι), Measurable (u i))
Goal: Measurable (Function.uncurry u)
-- MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable
theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι : Type*}
    [TopologicalSpace ι] [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι]
    [OpensMeasurableSpace ι] [TopologicalSpace β] [PseudoMetrizableSpace β] [MeasurableSpace α]
    {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x) (h : ∀ i, StronglyMeasurable (u i)) :
    StronglyMeasurable (Function.uncurry u) 
Arguments: {α : Type u_5} {β : Type u_6} {ι : Type u_7} [TopologicalSpace ι] [TopologicalSpace.MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [MeasurableSpace α] {u : ι → α → β} (hu_cont : ∀ (x : α), Continuous fun i => u i x) (h : ∀ (i : ι), MeasureTheory.StronglyMeasurable (u i))
Goal: MeasureTheory.StronglyMeasurable (Function.uncurry u)
-- Complex.measurableEquivPi_apply
@[simp]
theorem measurableEquivPi_apply (a : ℂ) :
    measurableEquivPi a = ![a.re, a.im] 
Arguments: (a : ℂ)
Goal: Complex.measurableEquivPi a = ![a.re, a.im]
-- Complex.measurableEquivPi_symm_apply
@[simp]
theorem measurableEquivPi_symm_apply (p : (Fin 2) → ℝ) :
    measurableEquivPi.symm p = (p 0) + (p 1) * I 
Arguments: (p : Fin 2 → ℝ)
Goal: Complex.measurableEquivPi.symm p = ↑(p 0) + ↑(p 1) * Complex.I
-- Complex.measurableEquivRealProd_apply
@[simp]
theorem measurableEquivRealProd_apply (a : ℂ) : measurableEquivRealProd a = (a.re, a.im) 
Arguments: (a : ℂ)
Goal: Complex.measurableEquivRealProd a = (a.re, a.im)
-- Complex.measurableEquivRealProd_symm_apply
@[simp]
theorem measurableEquivRealProd_symm_apply (p : ℝ × ℝ) :
    measurableEquivRealProd.symm p = {re := p.1, im := p.2} 
Arguments: (p : ℝ × ℝ)
Goal: Complex.measurableEquivRealProd.symm p = { re := p.1, im := p.2 }
-- Complex.volume_preserving_equiv_pi
theorem volume_preserving_equiv_pi : MeasurePreserving measurableEquivPi 
Goal: MeasureTheory.MeasurePreserving (⇑Complex.measurableEquivPi) MeasureTheory.volume MeasureTheory.volume
-- Complex.volume_preserving_equiv_real_prod
theorem volume_preserving_equiv_real_prod : MeasurePreserving measurableEquivRealProd 
Goal: MeasureTheory.MeasurePreserving (⇑Complex.measurableEquivRealProd) MeasureTheory.volume MeasureTheory.volume
-- MeasureTheory.AddContent.sUnion'
structure AddContent (C : Set (Set α)) where
    toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.AddContent C) (I : Finset (Set α)) (_h_ss : ↑I ⊆ C) (_h_dis : (↑I).PairwiseDisjoint id) (_h_mem : ⋃₀ ↑I ∈ C)
Goal: self.toFun (⋃₀ ↑I) = ∑ u ∈ I, self.toFun u
-- MeasureTheory.AddContent.empty'
structure AddContent (C : Set (Set α)) where
    toFun : Set α → ℝ≥0∞
  empty' : toFun ∅ = 0
  sUnion' (I : Finset (Set α)) (_h_ss : ↑I ⊆ C)
      (_h_dis : PairwiseDisjoint (I : Set (Set α)) id) (_h_mem : ⋃₀ ↑I ∈ C) :
    toFun (⋃₀ I) = ∑ u ∈ I, toFun u
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.AddContent C)
Goal: self.toFun ∅ = 0
-- MeasureTheory.AddContent.ext
@[ext] protected lemma AddContent.ext (h : ∀ s, m s = m' s) : m = m' 
Arguments: {α : Type u_1} {C : Set (Set α)} {m m' : MeasureTheory.AddContent C} (h : ∀ (s : Set α), m s = m' s)
Goal: m = m'
-- MeasureTheory.AddContent.ext_iff
protected lemma AddContent.ext_iff (m m' : AddContent C) : m = m' ↔ ∀ s, m s = m' s 
Arguments: {α : Type u_1} {C : Set (Set α)} (m m' : MeasureTheory.AddContent C)
Goal: m = m' ↔ ∀ (s : Set α), m s = m' s
-- MeasureTheory.addContent_empty
@[simp] lemma addContent_empty : m ∅ = 0 
Arguments: {α : Type u_1} {C : Set (Set α)} {m : MeasureTheory.AddContent C}
Goal: m ∅ = 0
-- MeasureTheory.addContent_sUnion
lemma addContent_sUnion (h_ss : ↑I ⊆ C)
    (h_dis : PairwiseDisjoint (I : Set (Set α)) id) (h_mem : ⋃₀ ↑I ∈ C) :
    m (⋃₀ I) = ∑ u ∈ I, m u 
Arguments: {α : Type u_1} {C : Set (Set α)} {I : Finset (Set α)} {m : MeasureTheory.AddContent C} (h_ss : ↑I ⊆ C) (h_dis : (↑I).PairwiseDisjoint id) (h_mem : ⋃₀ ↑I ∈ C)
Goal: m (⋃₀ ↑I) = ∑ u ∈ I, m u
-- MeasureTheory.addContent_union'
lemma addContent_union' (hs : s ∈ C) (ht : t ∈ C) (hst : s ∪ t ∈ C) (h_dis : Disjoint s t) :
    m (s ∪ t) = m s + m t 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} {m : MeasureTheory.AddContent C} (hs : s ∈ C) (ht : t ∈ C) (hst : s ∪ t ∈ C) (h_dis : Disjoint s t)
Goal: m (s ∪ t) = m s + m t
-- MeasureTheory.addContent_eq_add_diffFinset₀_of_subset
lemma addContent_eq_add_diffFinset₀_of_subset (hC : IsSetSemiring C)
    (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s)
    (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
    m s = ∑ i ∈ I, m i + ∑ i ∈ hC.diffFinset₀ hs hI, m i 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} {m : MeasureTheory.AddContent C} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) (h_dis : (↑I).PairwiseDisjoint id)
Goal: m s = ∑ i ∈ I, m i + ∑ i ∈ hC.diffFinset₀ hs hI, m i
-- MeasureTheory.sum_addContent_le_of_subset
lemma sum_addContent_le_of_subset (hC : IsSetSemiring C)
    (h_ss : ↑I ⊆ C) (h_dis : PairwiseDisjoint (I : Set (Set α)) id)
    (ht : t ∈ C) (hJt : ∀ s ∈ I, s ⊆ t) :
    ∑ u ∈ I, m u ≤ m t 
Arguments: {α : Type u_1} {C : Set (Set α)} {t : Set α} {I : Finset (Set α)} {m : MeasureTheory.AddContent C} (hC : MeasureTheory.IsSetSemiring C) (h_ss : ↑I ⊆ C) (h_dis : (↑I).PairwiseDisjoint id) (ht : t ∈ C) (hJt : ∀ s ∈ I, s ⊆ t)
Goal: ∑ u ∈ I, m u ≤ m t
-- MeasureTheory.addContent_mono
lemma addContent_mono (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
    (hst : s ⊆ t) :
    m s ≤ m t 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} {m : MeasureTheory.AddContent C} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) (hst : s ⊆ t)
Goal: m s ≤ m t
-- MeasureTheory.addContent_union
lemma addContent_union (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C)
    (h_dis : Disjoint s t) :
    m (s ∪ t) = m s + m t 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} {m : MeasureTheory.AddContent C} (hC : MeasureTheory.IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) (h_dis : Disjoint s t)
Goal: m (s ∪ t) = m s + m t
-- MeasureTheory.addContent_union_le
lemma addContent_union_le (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) :
    m (s ∪ t) ≤ m s + m t 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} {m : MeasureTheory.AddContent C} (hC : MeasureTheory.IsSetRing C) (hs : s ∈ C) (ht : t ∈ C)
Goal: m (s ∪ t) ≤ m s + m t
-- MeasureTheory.addContent_biUnion_le
lemma addContent_biUnion_le {ι : Type*} (hC : IsSetRing C) {s : ι → Set α}
    {S : Finset ι} (hs : ∀ n ∈ S, s n ∈ C) :
    m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i) 
Arguments: {α : Type u_1} {C : Set (Set α)} {m : MeasureTheory.AddContent C} {ι : Type u_2} (hC : MeasureTheory.IsSetRing C) {s : ι → Set α} {S : Finset ι} (hs : ∀ n ∈ S, s n ∈ C)
Goal: m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i)
-- MeasureTheory.le_addContent_diff
lemma le_addContent_diff (m : AddContent C) (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) :
    m s - m t ≤ m (s \ t) 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (m : MeasureTheory.AddContent C) (hC : MeasureTheory.IsSetRing C) (hs : s ∈ C) (ht : t ∈ C)
Goal: m s - m t ≤ m (s \ t)
-- MeasureTheory.ComplexMeasure.re_apply
@[simps! apply]
def re : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (v : MeasureTheory.VectorMeasure α ℂ)
Goal: MeasureTheory.ComplexMeasure.re v = v.mapRange Complex.reLm.toAddMonoidHom Complex.continuous_re
-- MeasureTheory.ComplexMeasure.im_apply
@[simps! apply]
def im : ComplexMeasure α →ₗ[ℝ] SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (v : MeasureTheory.VectorMeasure α ℂ)
Goal: MeasureTheory.ComplexMeasure.im v = v.mapRange Complex.imLm.toAddMonoidHom Complex.continuous_im
-- MeasureTheory.SignedMeasure.toComplexMeasure_apply_im
@[simps!]
def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (i : Set α)
Goal: (↑(s.toComplexMeasure t) i).im = ↑t i
-- MeasureTheory.SignedMeasure.toComplexMeasure_apply_re
@[simps!]
def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α) (i : Set α)
Goal: (↑(s.toComplexMeasure t) i).re = ↑s i
-- MeasureTheory.SignedMeasure.toComplexMeasure_apply
theorem _root_.MeasureTheory.SignedMeasure.toComplexMeasure_apply
    {s t : SignedMeasure α} {i : Set α} : s.toComplexMeasure t i = ⟨s i, t i⟩ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {s t : MeasureTheory.SignedMeasure α} {i : Set α}
Goal: ↑(s.toComplexMeasure t) i = { re := ↑s i, im := ↑t i }
-- MeasureTheory.ComplexMeasure.toComplexMeasure_to_signedMeasure
theorem toComplexMeasure_to_signedMeasure (c : ComplexMeasure α) :
    SignedMeasure.toComplexMeasure (ComplexMeasure.re c) (ComplexMeasure.im c) = c 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α)
Goal: (MeasureTheory.ComplexMeasure.re c).toComplexMeasure (MeasureTheory.ComplexMeasure.im c) = c
-- MeasureTheory.SignedMeasure.re_toComplexMeasure
theorem _root_.MeasureTheory.SignedMeasure.re_toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure.re (SignedMeasure.toComplexMeasure s t) = s 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α)
Goal: MeasureTheory.ComplexMeasure.re (s.toComplexMeasure t) = s
-- MeasureTheory.SignedMeasure.im_toComplexMeasure
theorem _root_.MeasureTheory.SignedMeasure.im_toComplexMeasure (s t : SignedMeasure α) :
    ComplexMeasure.im (SignedMeasure.toComplexMeasure s t) = t 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (s t : MeasureTheory.SignedMeasure α)
Goal: MeasureTheory.ComplexMeasure.im (s.toComplexMeasure t) = t
-- MeasureTheory.ComplexMeasure.equivSignedMeasure_apply
@[simps]
def equivSignedMeasure : ComplexMeasure α ≃ SignedMeasure α × SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α)
Goal: MeasureTheory.ComplexMeasure.equivSignedMeasure c =
  (MeasureTheory.ComplexMeasure.re c, MeasureTheory.ComplexMeasure.im c)
-- MeasureTheory.ComplexMeasure.equivSignedMeasure_symm_apply
@[simps]
def equivSignedMeasure : ComplexMeasure α ≃ SignedMeasure α × SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α}
Goal: ∀ (x : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasure.symm x =
    match x with
    | (s, t) => s.toComplexMeasure t
-- MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_symm_apply
@[simps]
def equivSignedMeasureₗ : ComplexMeasure α ≃ₗ[R] SignedMeasure α × SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {R : Type u_3} [Semiring R] [Module R ℝ] [ContinuousConstSMul R ℝ] [ContinuousConstSMul R ℂ]
Goal: ∀ (a : MeasureTheory.SignedMeasure α × MeasureTheory.SignedMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasureₗ.symm a = MeasureTheory.ComplexMeasure.equivSignedMeasure.invFun a
-- MeasureTheory.ComplexMeasure.equivSignedMeasureₗ_apply
@[simps]
def equivSignedMeasureₗ : ComplexMeasure α ≃ₗ[R] SignedMeasure α × SignedMeasure α 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {R : Type u_3} [Semiring R] [Module R ℝ] [ContinuousConstSMul R ℝ] [ContinuousConstSMul R ℂ]
Goal: ∀ (a : MeasureTheory.ComplexMeasure α),
  MeasureTheory.ComplexMeasure.equivSignedMeasureₗ a = MeasureTheory.ComplexMeasure.equivSignedMeasure.toFun a
-- MeasureTheory.ComplexMeasure.absolutelyContinuous_ennreal_iff
theorem absolutelyContinuous_ennreal_iff (c : ComplexMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
    c ≪ᵥ μ ↔ ComplexMeasure.re c ≪ᵥ μ ∧ ComplexMeasure.im c ≪ᵥ μ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} (c : MeasureTheory.ComplexMeasure α) (μ : MeasureTheory.VectorMeasure α ℝ≥0∞)
Goal: c ≪ᵥ μ ↔ MeasureTheory.ComplexMeasure.re c ≪ᵥ μ ∧ MeasureTheory.ComplexMeasure.im c ≪ᵥ μ
-- mem_parallelepiped_iff
theorem mem_parallelepiped_iff (v : ι → E) (x : E) :
    x ∈ parallelepiped v ↔ ∃ t ∈ Icc (0 : ι → ℝ) 1, x = ∑ i, t i • v i 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [AddCommGroup E] [Module ℝ E] (v : ι → E) (x : E)
Goal: x ∈ parallelepiped v ↔ ∃ t ∈ Set.Icc 0 1, x = ∑ i : ι, t i • v i
-- parallelepiped_basis_eq
theorem parallelepiped_basis_eq (b : Basis ι ℝ E) :
    parallelepiped b = {x | ∀ i, b.repr x i ∈ Set.Icc 0 1} 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [AddCommGroup E] [Module ℝ E] (b : Basis ι ℝ E)
Goal: parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Set.Icc 0 1}
-- image_parallelepiped
theorem image_parallelepiped (f : E →ₗ[ℝ] F) (v : ι → E) :
    f '' parallelepiped v = parallelepiped (f ∘ v) 
Arguments: {ι : Type u_1} {E : Type u_3} {F : Type u_4} [Fintype ι] [AddCommGroup E] [Module ℝ E] [AddCommGroup F] [Module ℝ F] (f : E →ₗ[ℝ] F) (v : ι → E)
Goal: ⇑f '' parallelepiped v = parallelepiped (⇑f ∘ v)
-- parallelepiped_comp_equiv
@[simp]
theorem parallelepiped_comp_equiv (v : ι → E) (e : ι' ≃ ι) :
    parallelepiped (v ∘ e) = parallelepiped v 
Arguments: {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} [Fintype ι] [Fintype ι'] [AddCommGroup E] [Module ℝ E] (v : ι → E) (e : ι' ≃ ι)
Goal: parallelepiped (v ∘ ⇑e) = parallelepiped v
-- parallelepiped_orthonormalBasis_one_dim
theorem parallelepiped_orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) :
    parallelepiped b = Icc 0 1 ∨ parallelepiped b = Icc (-1) 0 
Arguments: {ι : Type u_1} [Fintype ι] (b : OrthonormalBasis ι ℝ ℝ)
Goal: parallelepiped ⇑b = Set.Icc 0 1 ∨ parallelepiped ⇑b = Set.Icc (-1) 0
-- parallelepiped_eq_sum_segment
theorem parallelepiped_eq_sum_segment (v : ι → E) : parallelepiped v = ∑ i, segment ℝ 0 (v i) 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [AddCommGroup E] [Module ℝ E] (v : ι → E)
Goal: parallelepiped v = ∑ i : ι, segment ℝ 0 (v i)
-- convex_parallelepiped
theorem convex_parallelepiped (v : ι → E) : Convex ℝ (parallelepiped v) 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [AddCommGroup E] [Module ℝ E] (v : ι → E)
Goal: Convex ℝ (parallelepiped v)
-- parallelepiped_eq_convexHull
theorem parallelepiped_eq_convexHull (v : ι → E) :
    parallelepiped v = convexHull ℝ (∑ i, {(0 : E), v i}) 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [AddCommGroup E] [Module ℝ E] (v : ι → E)
Goal: parallelepiped v = (convexHull ℝ) (∑ i : ι, {0, v i})
-- parallelepiped_single
theorem parallelepiped_single [DecidableEq ι] (a : ι → ℝ) :
    (parallelepiped fun i => Pi.single i (a i)) = Set.uIcc 0 a 
Arguments: {ι : Type u_1} [Fintype ι] [DecidableEq ι] (a : ι → ℝ)
Goal: (parallelepiped fun i => Pi.single i (a i)) = Set.uIcc 0 a
-- Basis.coe_parallelepiped
@[simp]
theorem Basis.coe_parallelepiped (b : Basis ι ℝ E) :
    (b.parallelepiped : Set E) = _root_.parallelepiped b 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E)
Goal: ↑b.parallelepiped = parallelepiped ⇑b
-- Basis.parallelepiped_reindex
@[simp]
theorem Basis.parallelepiped_reindex (b : Basis ι ℝ E) (e : ι ≃ ι') :
    (b.reindex e).parallelepiped = b.parallelepiped 
Arguments: {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} [Fintype ι] [Fintype ι'] [NormedAddCommGroup E] [NormedSpace ℝ E] (b : Basis ι ℝ E) (e : ι ≃ ι')
Goal: (b.reindex e).parallelepiped = b.parallelepiped
-- Basis.parallelepiped_map
theorem Basis.parallelepiped_map (b : Basis ι ℝ E) (e : E ≃ₗ[ℝ] F) :
    (b.map e).parallelepiped = b.parallelepiped.map e
    (have := FiniteDimensional.of_fintype_basis b
    -- Porting note: Lean cannot infer the instance above
    LinearMap.continuous_of_finiteDimensional e.toLinearMap)
    (have := FiniteDimensional.of_fintype_basis (b.map e)
    -- Porting note: Lean cannot infer the instance above
    LinearMap.isOpenMap_of_finiteDimensional _ e.surjective) 
Arguments: {ι : Type u_1} {E : Type u_3} {F : Type u_4} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] (b : Basis ι ℝ E) (e : E ≃ₗ[ℝ] F)
Goal: (b.map e).parallelepiped = TopologicalSpace.PositiveCompacts.map ⇑e ⋯ ⋯ b.parallelepiped
-- Basis.prod_parallelepiped
theorem Basis.prod_parallelepiped (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
    (v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped 
Arguments: {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} {F : Type u_4} [Fintype ι] [Fintype ι'] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] (v : Basis ι ℝ E) (w : Basis ι' ℝ F)
Goal: (v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped
-- Basis.addHaar_def
irreducible_def Basis.addHaar (b : Basis ι ℝ E) : Measure E 
Arguments: {ι : Type u_5} {E : Type u_6} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (b : Basis ι ℝ E)
Goal: b.addHaar = MeasureTheory.Measure.addHaarMeasure b.parallelepiped
-- Basis.addHaar_eq_iff
theorem Basis.addHaar_eq_iff [SecondCountableTopology E] (b : Basis ι ℝ E) (μ : Measure E)
    [SigmaFinite μ] [IsAddLeftInvariant μ] :
    b.addHaar = μ ↔ μ b.parallelepiped = 1 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] (b : Basis ι ℝ E) (μ : MeasureTheory.Measure E) [MeasureTheory.SigmaFinite μ] [μ.IsAddLeftInvariant]
Goal: b.addHaar = μ ↔ μ ↑b.parallelepiped = 1
-- Basis.addHaar_reindex
@[simp]
theorem Basis.addHaar_reindex (b : Basis ι ℝ E) (e : ι ≃ ι') :
    (b.reindex e).addHaar = b.addHaar 
Arguments: {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} [Fintype ι] [Fintype ι'] [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (b : Basis ι ℝ E) (e : ι ≃ ι')
Goal: (b.reindex e).addHaar = b.addHaar
-- Basis.addHaar_self
theorem Basis.addHaar_self (b : Basis ι ℝ E) : b.addHaar (_root_.parallelepiped b) = 1 
Arguments: {ι : Type u_1} {E : Type u_3} [Fintype ι] [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] (b : Basis ι ℝ E)
Goal: b.addHaar (parallelepiped ⇑b) = 1
-- Basis.prod_addHaar
theorem Basis.prod_addHaar (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
    (v.prod w).addHaar = v.addHaar.prod w.addHaar 
Arguments: {ι : Type u_1} {ι' : Type u_2} {E : Type u_3} {F : Type u_4} [Fintype ι] [Fintype ι'] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] [SecondCountableTopologyEither E F] (v : Basis ι ℝ E) (w : Basis ι' ℝ F)
Goal: (v.prod w).addHaar = v.addHaar.prod w.addHaar
-- EuclideanSpace.measurableEquiv_toEquiv
@[simps toEquiv]
protected def measurableEquiv : EuclideanSpace ℝ ι ≃ᵐ (ι → ℝ) 
Arguments: (ι : Type u_1)
Goal: (EuclideanSpace.measurableEquiv ι).toEquiv = WithLp.equiv 2 ((i : ι) → (fun x => ℝ) i)
-- EuclideanSpace.coe_measurableEquiv
theorem coe_measurableEquiv : ⇑(EuclideanSpace.measurableEquiv ι) = WithLp.equiv 2 _ 
Arguments: (ι : Type u_1)
Goal: ⇑(EuclideanSpace.measurableEquiv ι) = ⇑(WithLp.equiv 2 ((i : ι) → (fun x => ℝ) i))