ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.isTopologicalBasis_isOpen_lt_top
protected theorem Measure.isTopologicalBasis_isOpen_lt_top [TopologicalSpace α]
    (μ : Measure α) [IsLocallyFiniteMeasure μ] :
    TopologicalSpace.IsTopologicalBasis { s | IsOpen s ∧ μ s < ∞ } 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: TopologicalSpace.IsTopologicalBasis {s | IsOpen s ∧ μ s < ⊤}
-- MeasureTheory.IsFiniteMeasureOnCompacts.lt_top_of_isCompact
class IsFiniteMeasureOnCompacts [TopologicalSpace α] (μ : Measure α) : Prop where
  protected lt_top_of_isCompact : ∀ ⦃K : Set α⦄, IsCompact K → μ K < ∞
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] {μ : MeasureTheory.Measure α} [self : MeasureTheory.IsFiniteMeasureOnCompacts μ] ⦃K : Set α⦄
Goal: IsCompact K → μ K < ⊤
-- IsCompact.measure_lt_top
theorem _root_.IsCompact.measure_lt_top [TopologicalSpace α] {μ : Measure α}
    [IsFiniteMeasureOnCompacts μ] ⦃K : Set α⦄ (hK : IsCompact K) : μ K < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] ⦃K : Set α⦄ (hK : IsCompact K)
Goal: μ K < ⊤
-- IsCompact.measure_ne_top
theorem _root_.IsCompact.measure_ne_top [TopologicalSpace α] {μ : Measure α}
    [IsFiniteMeasureOnCompacts μ] ⦃K : Set α⦄ (hK : IsCompact K) : μ K ≠ ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] ⦃K : Set α⦄ (hK : IsCompact K)
Goal: μ K ≠ ⊤
-- Bornology.IsBounded.measure_lt_top
theorem _root_.Bornology.IsBounded.measure_lt_top [PseudoMetricSpace α] [ProperSpace α]
    {μ : Measure α} [IsFiniteMeasureOnCompacts μ] ⦃s : Set α⦄ (hs : Bornology.IsBounded s) :
    μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [PseudoMetricSpace α] [ProperSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] ⦃s : Set α⦄ (hs : Bornology.IsBounded s)
Goal: μ s < ⊤
-- MeasureTheory.measure_closedBall_lt_top
theorem measure_closedBall_lt_top [PseudoMetricSpace α] [ProperSpace α] {μ : Measure α}
    [IsFiniteMeasureOnCompacts μ] {x : α} {r : ℝ} : μ (Metric.closedBall x r) < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [PseudoMetricSpace α] [ProperSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {x : α} {r : ℝ}
Goal: μ (Metric.closedBall x r) < ⊤
-- MeasureTheory.measure_ball_lt_top
theorem measure_ball_lt_top [PseudoMetricSpace α] [ProperSpace α] {μ : Measure α}
    [IsFiniteMeasureOnCompacts μ] {x : α} {r : ℝ} : μ (Metric.ball x r) < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [PseudoMetricSpace α] [ProperSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasureOnCompacts μ] {x : α} {r : ℝ}
Goal: μ (Metric.ball x r) < ⊤
-- MeasureTheory.IsFiniteMeasureOnCompacts.smul
protected theorem IsFiniteMeasureOnCompacts.smul [TopologicalSpace α] (μ : Measure α)
    [IsFiniteMeasureOnCompacts μ] {c : ℝ≥0∞} (hc : c ≠ ∞) : IsFiniteMeasureOnCompacts (c • μ) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} [TopologicalSpace α] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasureOnCompacts μ] {c : ℝ≥0∞} (hc : c ≠ ⊤)
Goal: MeasureTheory.IsFiniteMeasureOnCompacts (c • μ)
-- MeasureTheory.exists_pos_measure_of_cover
theorem exists_pos_measure_of_cover [Countable ι] {U : ι → Set α} (hU : ⋃ i, U i = univ)
    (hμ : μ ≠ 0) : ∃ i, 0 < μ (U i) 
Arguments: {α : Type u_1} {ι : Type u_4} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [Countable ι] {U : ι → Set α} (hU : ⋃ i, U i = Set.univ) (hμ : μ ≠ 0)
Goal: ∃ i, 0 < μ (U i)
-- MeasureTheory.exists_pos_preimage_ball
theorem exists_pos_preimage_ball [PseudoMetricSpace δ] (f : α → δ) (x : δ) (hμ : μ ≠ 0) :
    ∃ n : ℕ, 0 < μ (f ⁻¹' Metric.ball x n) 
Arguments: {α : Type u_1} {δ : Type u_3} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PseudoMetricSpace δ] (f : α → δ) (x : δ) (hμ : μ ≠ 0)
Goal: ∃ n, 0 < μ (f ⁻¹' Metric.ball x ↑n)
-- MeasureTheory.exists_pos_ball
theorem exists_pos_ball [PseudoMetricSpace α] (x : α) (hμ : μ ≠ 0) :
    ∃ n : ℕ, 0 < μ (Metric.ball x n) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [PseudoMetricSpace α] (x : α) (hμ : μ ≠ 0)
Goal: ∃ n, 0 < μ (Metric.ball x ↑n)
-- MeasureTheory.exists_ne_forall_mem_nhds_pos_measure_preimage
theorem exists_ne_forall_mem_nhds_pos_measure_preimage {β} [TopologicalSpace β] [T1Space β]
    [SecondCountableTopology β] [Nonempty β] {f : α → β} (h : ∀ b, ∃ᵐ x ∂μ, f x ≠ b) :
    ∃ a b : β, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {β : Type u_5} [TopologicalSpace β] [T1Space β] [SecondCountableTopology β] [Nonempty β] {f : α → β} (h : ∀ (b : β), ∃ᵐ (x : α) ∂μ, f x ≠ b)
Goal: ∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)
-- MeasureTheory.ext_on_measurableSpace_of_generate_finite
theorem ext_on_measurableSpace_of_generate_finite {α} (m₀ : MeasurableSpace α) {μ ν : Measure α}
    [IsFiniteMeasure μ] (C : Set (Set α)) (hμν : ∀ s ∈ C, μ s = ν s) {m : MeasurableSpace α}
    (h : m ≤ m₀) (hA : m = MeasurableSpace.generateFrom C) (hC : IsPiSystem C)
    (h_univ : μ Set.univ = ν Set.univ) {s : Set α} (hs : MeasurableSet[m] s) : μ s = ν s 
Arguments: {α : Type u_5} (m₀ : MeasurableSpace α) {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] (C : Set (Set α)) (hμν : ∀ s ∈ C, μ s = ν s) {m : MeasurableSpace α} (h : m ≤ m₀) (hA : m = MeasurableSpace.generateFrom C) (hC : IsPiSystem C) (h_univ : μ Set.univ = ν Set.univ) {s : Set α} (hs : MeasurableSet s)
Goal: μ s = ν s
-- MeasureTheory.ext_of_generate_finite
theorem ext_of_generate_finite (C : Set (Set α)) (hA : m0 = generateFrom C) (hC : IsPiSystem C)
    [IsFiniteMeasure μ] (hμν : ∀ s ∈ C, μ s = ν s) (h_univ : μ univ = ν univ) : μ = ν 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} (C : Set (Set α)) (hA : m0 = MeasurableSpace.generateFrom C) (hC : IsPiSystem C) [MeasureTheory.IsFiniteMeasure μ] (hμν : ∀ s ∈ C, μ s = ν s) (h_univ : μ Set.univ = ν Set.univ)
Goal: μ = ν
-- MeasureTheory.Measure.FiniteSpanningSetsIn.disjointed_set_eq
theorem FiniteSpanningSetsIn.disjointed_set_eq {μ : Measure α}
    (S : μ.FiniteSpanningSetsIn { s | MeasurableSet s }) : S.disjointed.set = disjointed S.set 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} (S : μ.FiniteSpanningSetsIn {s | MeasurableSet s})
Goal: S.disjointed.set = disjointed S.set
-- MeasureTheory.Measure.exists_eq_disjoint_finiteSpanningSetsIn
theorem exists_eq_disjoint_finiteSpanningSetsIn (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν] :
    ∃ (S : μ.FiniteSpanningSetsIn { s | MeasurableSet s })
      (T : ν.FiniteSpanningSetsIn { s | MeasurableSet s }),
      S.set = T.set ∧ Pairwise (Disjoint on S.set) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
Goal: ∃ S T, S.set = T.set ∧ Pairwise (Disjoint on S.set)
-- MeasureTheory.Measure.FiniteAtFilter.filter_mono
theorem filter_mono (h : f ≤ g) : μ.FiniteAtFilter g → μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : Filter α} (h : f ≤ g)
Goal: μ.FiniteAtFilter g → μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.inf_of_left
theorem inf_of_left (h : μ.FiniteAtFilter f) : μ.FiniteAtFilter (f ⊓ g) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : Filter α} (h : μ.FiniteAtFilter f)
Goal: μ.FiniteAtFilter (f ⊓ g)
-- MeasureTheory.Measure.FiniteAtFilter.inf_of_right
theorem inf_of_right (h : μ.FiniteAtFilter g) : μ.FiniteAtFilter (f ⊓ g) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : Filter α} (h : μ.FiniteAtFilter g)
Goal: μ.FiniteAtFilter (f ⊓ g)
-- MeasureTheory.Measure.FiniteAtFilter.inf_ae_iff
@[simp]
theorem inf_ae_iff : μ.FiniteAtFilter (f ⊓ ae μ) ↔ μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : Filter α}
Goal: μ.FiniteAtFilter (f ⊓ MeasureTheory.ae μ) ↔ μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.filter_mono_ae
theorem filter_mono_ae (h : f ⊓ (ae μ) ≤ g) (hg : μ.FiniteAtFilter g) : μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : Filter α} (h : f ⊓ MeasureTheory.ae μ ≤ g) (hg : μ.FiniteAtFilter g)
Goal: μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.measure_mono
protected theorem measure_mono (h : μ ≤ ν) : ν.FiniteAtFilter f → μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : Filter α} (h : μ ≤ ν)
Goal: ν.FiniteAtFilter f → μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.mono
@[mono]
protected theorem mono (hf : f ≤ g) (hμ : μ ≤ ν) : ν.FiniteAtFilter g → μ.FiniteAtFilter f 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f g : Filter α} (hf : f ≤ g) (hμ : μ ≤ ν)
Goal: ν.FiniteAtFilter g → μ.FiniteAtFilter f
-- MeasureTheory.Measure.FiniteAtFilter.eventually
protected theorem eventually (h : μ.FiniteAtFilter f) : ∀ᶠ s in f.smallSets, μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : Filter α} (h : μ.FiniteAtFilter f)
Goal: ∀ᶠ (s : Set α) in f.smallSets, μ s < ⊤
-- MeasureTheory.Measure.FiniteAtFilter.filterSup
theorem filterSup : μ.FiniteAtFilter f → μ.FiniteAtFilter g → μ.FiniteAtFilter (f ⊔ g) 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f g : Filter α}
Goal: μ.FiniteAtFilter f → μ.FiniteAtFilter g → μ.FiniteAtFilter (f ⊔ g)
-- MeasureTheory.Measure.finiteAt_nhdsWithin
theorem finiteAt_nhdsWithin [TopologicalSpace α] {_m0 : MeasurableSpace α} (μ : Measure α)
    [IsLocallyFiniteMeasure μ] (x : α) (s : Set α) : μ.FiniteAtFilter (𝓝[s] x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {_m0 : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ] (x : α) (s : Set α)
Goal: μ.FiniteAtFilter (𝓝[s] x)
-- MeasureTheory.Measure.finiteAt_principal
@[simp]
theorem finiteAt_principal : μ.FiniteAtFilter (𝓟 s) ↔ μ s < ∞ 
Arguments: {α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}
Goal: μ.FiniteAtFilter (𝓟 s) ↔ μ s < ⊤
-- MeasureTheory.Measure.isLocallyFiniteMeasure_of_le
theorem isLocallyFiniteMeasure_of_le [TopologicalSpace α] {_m : MeasurableSpace α} {μ ν : Measure α}
    [H : IsLocallyFiniteMeasure μ] (h : ν ≤ μ) : IsLocallyFiniteMeasure ν 
Arguments: {α : Type u_1} [TopologicalSpace α] {_m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [H : MeasureTheory.IsLocallyFiniteMeasure μ] (h : ν ≤ μ)
Goal: MeasureTheory.IsLocallyFiniteMeasure ν
-- IsCompact.exists_open_superset_measure_lt_top'
theorem exists_open_superset_measure_lt_top' (h : IsCompact s)
    (hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x)) : ∃ U ⊇ s, IsOpen U ∧ μ U < ∞ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h : IsCompact s) (hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝 x))
Goal: ∃ U ⊇ s, IsOpen U ∧ μ U < ⊤
-- IsCompact.exists_open_superset_measure_lt_top
theorem exists_open_superset_measure_lt_top (h : IsCompact s) (μ : Measure α)
    [IsLocallyFiniteMeasure μ] : ∃ U ⊇ s, IsOpen U ∧ μ U < ∞ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {s : Set α} (h : IsCompact s) (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: ∃ U ⊇ s, IsOpen U ∧ μ U < ⊤
-- IsCompact.measure_lt_top_of_nhdsWithin
theorem measure_lt_top_of_nhdsWithin (h : IsCompact s) (hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝[s] x)) :
    μ s < ∞ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (h : IsCompact s) (hμ : ∀ x ∈ s, μ.FiniteAtFilter (𝓝[s] x))
Goal: μ s < ⊤
-- IsCompact.measure_zero_of_nhdsWithin
theorem measure_zero_of_nhdsWithin (hs : IsCompact s) :
    (∀ a ∈ s, ∃ t ∈ 𝓝[s] a, μ t = 0) → μ s = 0 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} {s : Set α} (hs : IsCompact s)
Goal: (∀ a ∈ s, ∃ t ∈ 𝓝[s] a, μ t = 0) → μ s = 0
-- isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace
theorem isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace [TopologicalSpace α]
    [MeasurableSpace α] {μ : Measure α} [CompactSpace α] :
    IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [CompactSpace α]
Goal: MeasureTheory.IsFiniteMeasure μ ↔ MeasureTheory.IsFiniteMeasureOnCompacts μ
-- MeasureTheory.Measure.finiteSpanningSetsInOpen'_def
noncomputable irreducible_def MeasureTheory.Measure.finiteSpanningSetsInOpen' [TopologicalSpace α]
  [SecondCountableTopology α] {m : MeasurableSpace α} (μ : Measure α) [IsLocallyFiniteMeasure μ] :
  μ.FiniteSpanningSetsIn { K | IsOpen K } 
Arguments: {α : Type u_5} [TopologicalSpace α] [SecondCountableTopology α] {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsLocallyFiniteMeasure μ]
Goal: μ.finiteSpanningSetsInOpen' =
  let_fun H := ⋯;
  H.some
-- measure_Icc_lt_top
theorem measure_Icc_lt_top : μ (Icc a b) < ∞ 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a b : α}
Goal: μ (Set.Icc a b) < ⊤
-- measure_Ico_lt_top
theorem measure_Ico_lt_top : μ (Ico a b) < ∞ 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a b : α}
Goal: μ (Set.Ico a b) < ⊤
-- measure_Ioc_lt_top
theorem measure_Ioc_lt_top : μ (Ioc a b) < ∞ 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a b : α}
Goal: μ (Set.Ioc a b) < ⊤
-- measure_Ioo_lt_top
theorem measure_Ioo_lt_top : μ (Ioo a b) < ∞ 
Arguments: {α : Type u_1} [Preorder α] [TopologicalSpace α] [CompactIccSpace α] {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a b : α}
Goal: μ (Set.Ioo a b) < ⊤
-- WStarAlgebra.exists_predual
class WStarAlgebra (M : Type u) [NormedRing M] [StarRing M] [CstarRing M] [Module ℂ M]
    [NormedAlgebra ℂ M] [StarModule ℂ M] : Prop where
    exists_predual :
    ∃ (X : Type u) (_ : NormedAddCommGroup X) (_ : NormedSpace ℂ X) (_ : CompleteSpace X),
      Nonempty (NormedSpace.Dual ℂ X ≃ₗᵢ⋆[ℂ] M)
Arguments: {M : Type u} [NormedRing M] [StarRing M] [CstarRing M] [Module ℂ M] [NormedAlgebra ℂ M] [StarModule ℂ M] [self : WStarAlgebra M]
Goal: ∃ X x x_1, ∃ (_ : CompleteSpace X), Nonempty (NormedSpace.Dual ℂ X ≃ₗᵢ⋆[ℂ] M)
-- VonNeumannAlgebra.centralizer_centralizer'
-- Porting note: I don't think the nonempty instance linter exists yet
structure VonNeumannAlgebra (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]
    [CompleteSpace H] extends StarSubalgebra ℂ (H →L[ℂ] H) where
    centralizer_centralizer' : Set.centralizer (Set.centralizer carrier) = carrier
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (self : VonNeumannAlgebra H)
Goal: self.carrier.centralizer.centralizer = self.carrier
-- VonNeumannAlgebra.mem_carrier
@[simp]
theorem mem_carrier {S : VonNeumannAlgebra H} {x : H →L[ℂ] H} :
    x ∈ S.toStarSubalgebra ↔ x ∈ (S : Set (H →L[ℂ] H)) 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] {S : VonNeumannAlgebra H} {x : H →L[ℂ] H}
Goal: x ∈ S.toStarSubalgebra ↔ x ∈ ↑S
-- VonNeumannAlgebra.coe_toStarSubalgebra
@[simp]
theorem coe_toStarSubalgebra (S : VonNeumannAlgebra H) :
    (S.toStarSubalgebra : Set (H →L[ℂ] H)) = S 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : VonNeumannAlgebra H)
Goal: ↑S.toStarSubalgebra = ↑S
-- VonNeumannAlgebra.coe_mk
@[simp]
theorem coe_mk (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h) :
    ((⟨S, h⟩ : VonNeumannAlgebra H) : Set (H →L[ℂ] H)) = S 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) (h : S.carrier.centralizer.centralizer = S.carrier)
Goal: ↑{ toStarSubalgebra := S, centralizer_centralizer' := h } = ↑S
-- VonNeumannAlgebra.ext
@[ext]
theorem ext {S T : VonNeumannAlgebra H} (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] {S T : VonNeumannAlgebra H} (h : ∀ (x : H →L[ℂ] H), x ∈ S ↔ x ∈ T)
Goal: S = T
-- VonNeumannAlgebra.centralizer_centralizer
@[simp]
theorem centralizer_centralizer (S : VonNeumannAlgebra H) :
    Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : VonNeumannAlgebra H)
Goal: (↑S).centralizer.centralizer = ↑S
-- VonNeumannAlgebra.coe_commutant
@[simp]
theorem coe_commutant (S : VonNeumannAlgebra H) :
    ↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : VonNeumannAlgebra H)
Goal: ↑S.commutant = (↑S).centralizer
-- VonNeumannAlgebra.mem_commutant_iff
@[simp]
theorem mem_commutant_iff {S : VonNeumannAlgebra H} {z : H →L[ℂ] H} :
    z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] {S : VonNeumannAlgebra H} {z : H →L[ℂ] H}
Goal: z ∈ S.commutant ↔ ∀ g ∈ S, g * z = z * g
-- VonNeumannAlgebra.commutant_commutant
@[simp]
theorem commutant_commutant (S : VonNeumannAlgebra H) : S.commutant.commutant = S 
Arguments: {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : VonNeumannAlgebra H)
Goal: S.commutant.commutant = S
-- MeasureTheory.FiniteMeasure.val_eq_toMeasure
@[simp]
theorem val_eq_toMeasure (ν : FiniteMeasure Ω) : ν.val = (ν : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.FiniteMeasure Ω)
Goal: ↑ν = ↑ν
-- MeasureTheory.FiniteMeasure.toMeasure_injective
theorem toMeasure_injective : Function.Injective ((↑) : FiniteMeasure Ω → Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: Function.Injective MeasureTheory.FiniteMeasure.toMeasure
-- MeasureTheory.FiniteMeasure.coeFn_def
lemma coeFn_def (μ : FiniteMeasure Ω) : μ = fun s ↦ ((μ : Measure Ω) s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: ⇑μ = fun s => (↑μ s).toNNReal
-- MeasureTheory.FiniteMeasure.coeFn_mk
lemma coeFn_mk (μ : Measure Ω) (hμ) :
    DFunLike.coe (F := FiniteMeasure Ω) ⟨μ, hμ⟩ = fun s ↦ (μ s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (hμ : MeasureTheory.IsFiniteMeasure μ)
Goal: ⇑⟨μ, hμ⟩ = fun s => (μ s).toNNReal
-- MeasureTheory.FiniteMeasure.mk_apply
@[simp, norm_cast]
lemma mk_apply (μ : Measure Ω) (hμ) (s : Set Ω) :
    DFunLike.coe (F := FiniteMeasure Ω) ⟨μ, hμ⟩ s = (μ s).toNNReal 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) (hμ : MeasureTheory.IsFiniteMeasure μ) (s : Set Ω)
Goal: ⟨μ, hμ⟩ s = (μ s).toNNReal
-- MeasureTheory.FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure
@[simp]
theorem ennreal_coeFn_eq_coeFn_toMeasure (ν : FiniteMeasure Ω) (s : Set Ω) :
    (ν s : ℝ≥0∞) = (ν : Measure Ω) s 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (ν : MeasureTheory.FiniteMeasure Ω) (s : Set Ω)
Goal: ↑(ν s) = ↑ν s
-- MeasureTheory.FiniteMeasure.apply_mono
theorem apply_mono (μ : FiniteMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂)
Goal: μ s₁ ≤ μ s₂
-- MeasureTheory.FiniteMeasure.apply_le_mass
@[simp] theorem apply_le_mass (μ : FiniteMeasure Ω) (s : Set Ω) : μ s ≤ μ.mass 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω) (s : Set Ω)
Goal: μ s ≤ μ.mass
-- MeasureTheory.FiniteMeasure.ennreal_mass
@[simp]
theorem ennreal_mass {μ : FiniteMeasure Ω} : (μ.mass : ℝ≥0∞) = (μ : Measure Ω) univ 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.FiniteMeasure Ω}
Goal: ↑μ.mass = ↑μ Set.univ
-- MeasureTheory.FiniteMeasure.coeFn_zero
@[simp, norm_cast] lemma coeFn_zero : ⇑(0 : FiniteMeasure Ω) = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: ⇑0 = 0
-- MeasureTheory.FiniteMeasure.zero_mass
@[simp]
theorem zero_mass : (0 : FiniteMeasure Ω).mass = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: MeasureTheory.FiniteMeasure.mass 0 = 0
-- MeasureTheory.FiniteMeasure.mass_zero_iff
@[simp]
theorem mass_zero_iff (μ : FiniteMeasure Ω) : μ.mass = 0 ↔ μ = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: μ.mass = 0 ↔ μ = 0
-- MeasureTheory.FiniteMeasure.mass_nonzero_iff
theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.FiniteMeasure Ω)
Goal: μ.mass ≠ 0 ↔ μ ≠ 0
-- MeasureTheory.FiniteMeasure.eq_of_forall_toMeasure_apply_eq
@[ext]
theorem eq_of_forall_toMeasure_apply_eq (μ ν : FiniteMeasure Ω)
    (h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.FiniteMeasure Ω) (h : ∀ (s : Set Ω), MeasurableSet s → ↑μ s = ↑ν s)
Goal: μ = ν
-- MeasureTheory.FiniteMeasure.eq_of_forall_apply_eq
theorem eq_of_forall_apply_eq (μ ν : FiniteMeasure Ω)
    (h : ∀ s : Set Ω, MeasurableSet s → μ s = ν s) : μ = ν 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.FiniteMeasure Ω) (h : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s)
Goal: μ = ν
-- MeasureTheory.FiniteMeasure.toMeasure_zero
@[simp, norm_cast]
theorem toMeasure_zero : ((↑) : FiniteMeasure Ω → Measure Ω) 0 = 0 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: ↑0 = 0
-- MeasureTheory.FiniteMeasure.toMeasure_add
@[norm_cast]
theorem toMeasure_add (μ ν : FiniteMeasure Ω) : ↑(μ + ν) = (↑μ + ↑ν : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.FiniteMeasure Ω)
Goal: ↑(μ + ν) = ↑μ + ↑ν
-- MeasureTheory.FiniteMeasure.toMeasure_smul
@[simp, norm_cast]
theorem toMeasure_smul (c : R) (μ : FiniteMeasure Ω) : ↑(c • μ) = c • (μ : Measure Ω) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {R : Type u_2} [SMul R ℝ≥0] [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (c : R) (μ : MeasureTheory.FiniteMeasure Ω)
Goal: ↑(c • μ) = c • ↑μ
-- MeasureTheory.FiniteMeasure.coeFn_add
@[simp, norm_cast]
theorem coeFn_add (μ ν : FiniteMeasure Ω) : (⇑(μ + ν) : Set Ω → ℝ≥0) = (⇑μ + ⇑ν : Set Ω → ℝ≥0) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] (μ ν : MeasureTheory.FiniteMeasure Ω)
Goal: ⇑(μ + ν) = ⇑μ + ⇑ν
-- MeasureTheory.FiniteMeasure.coeFn_smul
@[simp, norm_cast]
theorem coeFn_smul [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : FiniteMeasure Ω) :
    (⇑(c • μ) : Set Ω → ℝ≥0) = c • (⇑μ : Set Ω → ℝ≥0) 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {R : Type u_2} [SMul R ℝ≥0] [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : MeasureTheory.FiniteMeasure Ω)
Goal: ⇑(c • μ) = c • ⇑μ
-- MeasureTheory.FiniteMeasure.toMeasureAddMonoidHom_apply
@[simps]
def toMeasureAddMonoidHom : FiniteMeasure Ω →+ Measure Ω 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω]
Goal: ∀ (a : MeasureTheory.FiniteMeasure Ω), MeasureTheory.FiniteMeasure.toMeasureAddMonoidHom a = ↑a
-- MeasureTheory.FiniteMeasure.smul_apply
@[simp]
theorem smul_apply [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : FiniteMeasure Ω) (s : Set Ω) :
    (c • μ) s = c • μ s 
Arguments: {Ω : Type u_1} [MeasurableSpace Ω] {R : Type u_2} [SMul R ℝ≥0] [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : MeasureTheory.FiniteMeasure Ω) (s : Set Ω)
Goal: (c • μ) s = c • μ s