ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- circleIntegral.integral_smul
@[simp]
theorem integral_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜)
    (f : ℂ → E) (c : ℂ) (R : ℝ) : (∮ z in C(c, R), a • f z) = a • ∮ z in C(c, R), f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {𝕜 : Type u_2} [RCLike 𝕜] [NormedSpace 𝕜 E] [SMulCommClass 𝕜 ℂ E] (a : 𝕜) (f : ℂ → E) (c : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), a • f z) = a • ∮ (z : ℂ) in C(c, R), f z
-- circleIntegral.integral_smul_const
@[simp]
theorem integral_smul_const (f : ℂ → ℂ) (a : E) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), f z • a) = (∮ z in C(c, R), f z) • a 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] (f : ℂ → ℂ) (a : E) (c : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), f z • a) = (∮ (z : ℂ) in C(c, R), f z) • a
-- circleIntegral.integral_const_mul
@[simp]
theorem integral_const_mul (a : ℂ) (f : ℂ → ℂ) (c : ℂ) (R : ℝ) :
    (∮ z in C(c, R), a * f z) = a * ∮ z in C(c, R), f z 
Arguments: (a : ℂ) (f : ℂ → ℂ) (c : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), a * f z) = a * ∮ (z : ℂ) in C(c, R), f z
-- circleIntegral.integral_sub_center_inv
@[simp]
theorem integral_sub_center_inv (c : ℂ) {R : ℝ} (hR : R ≠ 0) :
    (∮ z in C(c, R), (z - c)⁻¹) = 2 * π * I 
Arguments: (c : ℂ) {R : ℝ} (hR : R ≠ 0)
Goal: (∮ (z : ℂ) in C(c, R), (z - c)⁻¹) = 2 * ↑π * Complex.I
-- circleIntegral.integral_eq_zero_of_hasDerivWithinAt'
theorem integral_eq_zero_of_hasDerivWithinAt' {f f' : ℂ → E} {c : ℂ} {R : ℝ}
    (h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z) :
    (∮ z in C(c, R), f' z) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f f' : ℂ → E} {c : ℂ} {R : ℝ} (h : ∀ z ∈ Metric.sphere c |R|, HasDerivWithinAt f (f' z) (Metric.sphere c |R|) z)
Goal: (∮ (z : ℂ) in C(c, R), f' z) = 0
-- circleIntegral.integral_eq_zero_of_hasDerivWithinAt
theorem integral_eq_zero_of_hasDerivWithinAt {f f' : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R)
    (h : ∀ z ∈ sphere c R, HasDerivWithinAt f (f' z) (sphere c R) z) : (∮ z in C(c, R), f' z) = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f f' : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R) (h : ∀ z ∈ Metric.sphere c R, HasDerivWithinAt f (f' z) (Metric.sphere c R) z)
Goal: (∮ (z : ℂ) in C(c, R), f' z) = 0
-- circleIntegral.integral_sub_zpow_of_undef
theorem integral_sub_zpow_of_undef {n : ℤ} {c w : ℂ} {R : ℝ} (hn : n < 0)
    (hw : w ∈ sphere c |R|) : (∮ z in C(c, R), (z - w) ^ n) = 0 
Arguments: {n : ℤ} {c w : ℂ} {R : ℝ} (hn : n < 0) (hw : w ∈ Metric.sphere c |R|)
Goal: (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0
-- circleIntegral.integral_sub_zpow_of_ne
theorem integral_sub_zpow_of_ne {n : ℤ} (hn : n ≠ -1) (c w : ℂ) (R : ℝ) :
    (∮ z in C(c, R), (z - w) ^ n) = 0 
Arguments: {n : ℤ} (hn : n ≠ -1) (c w : ℂ) (R : ℝ)
Goal: (∮ (z : ℂ) in C(c, R), (z - w) ^ n) = 0
-- cauchyPowerSeries_apply
theorem cauchyPowerSeries_apply (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) (w : ℂ) :
    (cauchyPowerSeries f c R n fun _ => w) =
      (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) (w : ℂ)
Goal: ((cauchyPowerSeries f c R n) fun x => w) =
  (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z
-- norm_cauchyPowerSeries_le
theorem norm_cauchyPowerSeries_le (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ) :
    ‖cauchyPowerSeries f c R n‖ ≤
      ((2 * π)⁻¹ * ∫ θ : ℝ in (0)..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c : ℂ) (R : ℝ) (n : ℕ)
Goal: ‖cauchyPowerSeries f c R n‖ ≤ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0 ..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n
-- le_radius_cauchyPowerSeries
theorem le_radius_cauchyPowerSeries (f : ℂ → E) (c : ℂ) (R : ℝ≥0) :
    ↑R ≤ (cauchyPowerSeries f c R).radius 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (c : ℂ) (R : ℝ≥0)
Goal: ↑R ≤ (cauchyPowerSeries f c ↑R).radius
-- hasSum_two_pi_I_cauchyPowerSeries_integral
theorem hasSum_two_pi_I_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    HasSum (fun n : ℕ => ∮ z in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z)
      (∮ z in C(c, R), (z - (c + w))⁻¹ • f z) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ} (hf : CircleIntegrable f c R) (hw : Complex.abs w < R)
Goal: HasSum (fun n => ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z)
  (∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z)
-- hasSum_cauchyPowerSeries_integral
theorem hasSum_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    HasSum (fun n => cauchyPowerSeries f c R n fun _ => w)
      ((2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - (c + w))⁻¹ • f z) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ} (hf : CircleIntegrable f c R) (hw : Complex.abs w < R)
Goal: HasSum (fun n => (cauchyPowerSeries f c R n) fun x => w)
  ((2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z)
-- sum_cauchyPowerSeries_eq_integral
theorem sum_cauchyPowerSeries_eq_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
    (hf : CircleIntegrable f c R) (hw : abs w < R) :
    (cauchyPowerSeries f c R).sum w = (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - (c + w))⁻¹ • f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ} (hf : CircleIntegrable f c R) (hw : Complex.abs w < R)
Goal: (cauchyPowerSeries f c R).sum w = (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - (c + w))⁻¹ • f z
-- hasFPowerSeriesOn_cauchy_integral
theorem hasFPowerSeriesOn_cauchy_integral {f : ℂ → E} {c : ℂ} {R : ℝ≥0}
    (hf : CircleIntegrable f c R) (hR : 0 < R) :
    HasFPowerSeriesOnBall (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z)
      (cauchyPowerSeries f c R) c R 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} {c : ℂ} {R : ℝ≥0} (hf : CircleIntegrable f c ↑R) (hR : 0 < R)
Goal: HasFPowerSeriesOnBall (fun w => (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z)
  (cauchyPowerSeries f c ↑R) c ↑R
-- circleIntegral.integral_sub_inv_of_mem_ball
theorem integral_sub_inv_of_mem_ball {c w : ℂ} {R : ℝ} (hw : w ∈ ball c R) :
    (∮ z in C(c, R), (z - w)⁻¹) = 2 * π * I 
Arguments: {c w : ℂ} {R : ℝ} (hw : w ∈ Metric.ball c R)
Goal: (∮ (z : ℂ) in C(c, R), (z - w)⁻¹) = 2 * ↑π * Complex.I
-- borel_anti
theorem borel_anti : Antitone (@borel α) 
Arguments: {α : Type u_1}
Goal: Antitone (@borel α)
-- borel_eq_top_of_discrete
theorem borel_eq_top_of_discrete [TopologicalSpace α] [DiscreteTopology α] : borel α = ⊤ 
Arguments: {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α]
Goal: borel α = ⊤
-- borel_eq_top_of_countable
theorem borel_eq_top_of_countable [TopologicalSpace α] [T1Space α] [Countable α] : borel α = ⊤ 
Arguments: {α : Type u_1} [TopologicalSpace α] [T1Space α] [Countable α]
Goal: borel α = ⊤
-- borel_eq_generateFrom_of_subbasis
theorem borel_eq_generateFrom_of_subbasis {s : Set (Set α)} [t : TopologicalSpace α]
    [SecondCountableTopology α] (hs : t = .generateFrom s) : borel α = .generateFrom s 
Arguments: {α : Type u_1} {s : Set (Set α)} [t : TopologicalSpace α] [SecondCountableTopology α] (hs : t = TopologicalSpace.generateFrom s)
Goal: borel α = MeasurableSpace.generateFrom s
-- TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom
theorem TopologicalSpace.IsTopologicalBasis.borel_eq_generateFrom [TopologicalSpace α]
    [SecondCountableTopology α] {s : Set (Set α)} (hs : IsTopologicalBasis s) :
    borel α = .generateFrom s 
Arguments: {α : Type u_1} [TopologicalSpace α] [SecondCountableTopology α] {s : Set (Set α)} (hs : TopologicalSpace.IsTopologicalBasis s)
Goal: borel α = MeasurableSpace.generateFrom s
-- isPiSystem_isOpen
theorem isPiSystem_isOpen [TopologicalSpace α] : IsPiSystem ({s : Set α | IsOpen s}) 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: IsPiSystem {s | IsOpen s}
-- isPiSystem_isClosed
lemma isPiSystem_isClosed [TopologicalSpace α] : IsPiSystem ({s : Set α | IsClosed s}) 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: IsPiSystem {s | IsClosed s}
-- borel_eq_generateFrom_isClosed
theorem borel_eq_generateFrom_isClosed [TopologicalSpace α] :
    borel α = .generateFrom { s | IsClosed s } 
Arguments: {α : Type u_1} [TopologicalSpace α]
Goal: borel α = MeasurableSpace.generateFrom {s | IsClosed s}
-- borel_comap
theorem borel_comap {f : α → β} {t : TopologicalSpace β} :
    @borel α (t.induced f) = (@borel β t).comap f 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {t : TopologicalSpace β}
Goal: borel α = MeasurableSpace.comap f (borel β)
-- Continuous.borel_measurable
theorem Continuous.borel_measurable [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
    (hf : Continuous f) : @Measurable α β (borel α) (borel β) f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f)
Goal: Measurable f
-- OpensMeasurableSpace.borel_le
class OpensMeasurableSpace (α : Type*) [TopologicalSpace α] [h : MeasurableSpace α] : Prop where
    borel_le : borel α ≤ h
Arguments: {α : Type u_6} [TopologicalSpace α] [h : MeasurableSpace α] [self : OpensMeasurableSpace α]
Goal: borel α ≤ h
-- BorelSpace.measurable_eq
class BorelSpace (α : Type*) [TopologicalSpace α] [MeasurableSpace α] : Prop where
    measurable_eq : ‹MeasurableSpace α› = borel α
Arguments: {α : Type u_6} [TopologicalSpace α] [MeasurableSpace α] [self : BorelSpace α]
Goal: inst✝ = borel α
-- opensMeasurableSpace_iff_forall_measurableSet
lemma opensMeasurableSpace_iff_forall_measurableSet
    [TopologicalSpace α] [MeasurableSpace α] :
    OpensMeasurableSpace α ↔  (∀ (s : Set α), IsOpen s → MeasurableSet s) 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α]
Goal: OpensMeasurableSpace α ↔ ∀ (s : Set α), IsOpen s → MeasurableSet s
-- MeasurableSet.induction_on_open
theorem MeasurableSet.induction_on_open [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α]
    {C : Set α → Prop} (h_open : ∀ U, IsOpen U → C U)
    (h_compl : ∀ t, MeasurableSet t → C t → C tᶜ)
    (h_union :
      ∀ f : ℕ → Set α,
        Pairwise (Disjoint on f) → (∀ i, MeasurableSet (f i)) → (∀ i, C (f i)) → C (⋃ i, f i)) :
    ∀ ⦃t⦄, MeasurableSet t → C t 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {C : Set α → Prop} (h_open : ∀ (U : Set α), IsOpen U → C U) (h_compl : ∀ (t : Set α), MeasurableSet t → C t → C tᶜ) (h_union :
  ∀ (f : ℕ → Set α), Pairwise (Disjoint on f) → (∀ (i : ℕ), MeasurableSet (f i)) → (∀ (i : ℕ), C (f i)) → C (⋃ i, f i)) ⦃t : Set α⦄
Goal: MeasurableSet t → C t
-- IsOpen.measurableSet
theorem IsOpen.measurableSet (h : IsOpen s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsOpen s)
Goal: MeasurableSet s
-- IsOpen.nullMeasurableSet
theorem IsOpen.nullMeasurableSet {μ} (h : IsOpen s) : NullMeasurableSet s μ 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} (h : IsOpen s)
Goal: MeasureTheory.NullMeasurableSet s μ
-- measurableSet_interior
@[measurability]
theorem measurableSet_interior : MeasurableSet (interior s) 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
Goal: MeasurableSet (interior s)
-- IsGδ.measurableSet
theorem IsGδ.measurableSet (h : IsGδ s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsGδ s)
Goal: MeasurableSet s
-- measurableSet_of_continuousAt
theorem measurableSet_of_continuousAt {β} [EMetricSpace β] (f : α → β) :
    MeasurableSet { x | ContinuousAt f x } 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {β : Type u_6} [EMetricSpace β] (f : α → β)
Goal: MeasurableSet {x | ContinuousAt f x}
-- IsClosed.measurableSet
theorem IsClosed.measurableSet (h : IsClosed s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (h : IsClosed s)
Goal: MeasurableSet s
-- IsClosed.nullMeasurableSet
theorem IsClosed.nullMeasurableSet {μ} (h : IsClosed s) : NullMeasurableSet s μ 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} (h : IsClosed s)
Goal: MeasureTheory.NullMeasurableSet s μ
-- IsCompact.measurableSet
theorem IsCompact.measurableSet [T2Space α] (h : IsCompact s) : MeasurableSet s 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [T2Space α] (h : IsCompact s)
Goal: MeasurableSet s
-- Inseparable.mem_measurableSet_iff
theorem Inseparable.mem_measurableSet_iff {x y : γ} (h : Inseparable x y) {s : Set γ}
    (hs : MeasurableSet s) : x ∈ s ↔ y ∈ s 
Arguments: {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {x y : γ} (h : Inseparable x y) {s : Set γ} (hs : MeasurableSet s)
Goal: x ∈ s ↔ y ∈ s
-- IsCompact.closure_subset_measurableSet
theorem IsCompact.closure_subset_measurableSet [R1Space γ] {K s : Set γ} (hK : IsCompact K)
    (hs : MeasurableSet s) (hKs : K ⊆ s) : closure K ⊆ s 
Arguments: {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [R1Space γ] {K s : Set γ} (hK : IsCompact K) (hs : MeasurableSet s) (hKs : K ⊆ s)
Goal: closure K ⊆ s
-- IsCompact.measure_closure
theorem IsCompact.measure_closure [R1Space γ] {K : Set γ} (hK : IsCompact K) (μ : Measure γ) :
    μ (closure K) = μ K 
Arguments: {γ : Type u_3} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [R1Space γ] {K : Set γ} (hK : IsCompact K) (μ : MeasureTheory.Measure γ)
Goal: μ (closure K) = μ K
-- measurableSet_closure
@[measurability]
theorem measurableSet_closure : MeasurableSet (closure s) 
Arguments: {α : Type u_1} {s : Set α} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
Goal: MeasurableSet (closure s)
-- measurable_of_isOpen
theorem measurable_of_isOpen {f : δ → γ} (hf : ∀ s, IsOpen s → MeasurableSet (f ⁻¹' s)) :
    Measurable f 
Arguments: {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s))
Goal: Measurable f
-- measurable_of_isClosed
theorem measurable_of_isClosed {f : δ → γ} (hf : ∀ s, IsClosed s → MeasurableSet (f ⁻¹' s)) :
    Measurable f 
Arguments: {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsClosed s → MeasurableSet (f ⁻¹' s))
Goal: Measurable f
-- measurable_of_isClosed'
theorem measurable_of_isClosed' {f : δ → γ}
    (hf : ∀ s, IsClosed s → s.Nonempty → s ≠ univ → MeasurableSet (f ⁻¹' s)) : Measurable f 
Arguments: {γ : Type u_3} {δ : Type u_5} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] {f : δ → γ} (hf : ∀ (s : Set γ), IsClosed s → s.Nonempty → s ≠ Set.univ → MeasurableSet (f ⁻¹' s))
Goal: Measurable f
-- MeasurableSet.nhdsWithin_isMeasurablyGenerated
theorem MeasurableSet.nhdsWithin_isMeasurablyGenerated {s : Set α} (hs : MeasurableSet s) (a : α) :
    (𝓝[s] a).IsMeasurablyGenerated 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (hs : MeasurableSet s) (a : α)
Goal: (𝓝[s] a).IsMeasurablyGenerated
-- SecondCountableTopologyEither.out
class SecondCountableTopologyEither (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] :
  Prop where
    out : SecondCountableTopology α ∨ SecondCountableTopology β
Arguments: {α : Type u_6} {β : Type u_7} [TopologicalSpace α] [TopologicalSpace β] [self : SecondCountableTopologyEither α β]
Goal: SecondCountableTopology α ∨ SecondCountableTopology β
-- interior_ae_eq_of_null_frontier
theorem interior_ae_eq_of_null_frontier {μ : Measure α'} {s : Set α'} (h : μ (frontier s) = 0) :
    interior s =ᵐ[μ] s 
Arguments: {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : μ (frontier s) = 0)
Goal: interior s =ᵐ[μ] s
-- measure_interior_of_null_frontier
theorem measure_interior_of_null_frontier {μ : Measure α'} {s : Set α'} (h : μ (frontier s) = 0) :
    μ (interior s) = μ s 
Arguments: {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : μ (frontier s) = 0)
Goal: μ (interior s) = μ s
-- nullMeasurableSet_of_null_frontier
theorem nullMeasurableSet_of_null_frontier {s : Set α} {μ : Measure α} (h : μ (frontier s) = 0) :
    NullMeasurableSet s μ 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} {μ : MeasureTheory.Measure α} (h : μ (frontier s) = 0)
Goal: MeasureTheory.NullMeasurableSet s μ
-- closure_ae_eq_of_null_frontier
theorem closure_ae_eq_of_null_frontier {μ : Measure α'} {s : Set α'} (h : μ (frontier s) = 0) :
    closure s =ᵐ[μ] s 
Arguments: {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : μ (frontier s) = 0)
Goal: closure s =ᵐ[μ] s
-- measure_closure_of_null_frontier
theorem measure_closure_of_null_frontier {μ : Measure α'} {s : Set α'} (h : μ (frontier s) = 0) :
    μ (closure s) = μ s 
Arguments: {α' : Type u_6} [TopologicalSpace α'] [MeasurableSpace α'] {μ : MeasureTheory.Measure α'} {s : Set α'} (h : μ (frontier s) = 0)
Goal: μ (closure s) = μ s
-- Continuous.measurable
@[fun_prop]
theorem Continuous.measurable {f : α → γ} (hf : Continuous f) : Measurable f 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (hf : Continuous f)
Goal: Measurable f
-- Continuous.aemeasurable
theorem Continuous.aemeasurable {f : α → γ} (h : Continuous f) {μ : Measure α} : AEMeasurable f μ 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (h : Continuous f) {μ : MeasureTheory.Measure α}
Goal: AEMeasurable f μ
-- ClosedEmbedding.measurable
theorem ClosedEmbedding.measurable {f : α → γ} (hf : ClosedEmbedding f) : Measurable f 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f : α → γ} (hf : ClosedEmbedding f)
Goal: Measurable f
-- ContinuousOn.measurable_piecewise
theorem ContinuousOn.measurable_piecewise {f g : α → γ} {s : Set α} [∀ j : α, Decidable (j ∈ s)]
    (hf : ContinuousOn f s) (hg : ContinuousOn g sᶜ) (hs : MeasurableSet s) :
    Measurable (s.piecewise f g) 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] {f g : α → γ} {s : Set α} [(j : α) → Decidable (j ∈ s)] (hf : ContinuousOn f s) (hg : ContinuousOn g sᶜ) (hs : MeasurableSet s)
Goal: Measurable (s.piecewise f g)
-- Homeomorph.measurable
@[measurability]
protected theorem Homeomorph.measurable (h : α ≃ₜ γ) : Measurable h 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] (h : α ≃ₜ γ)
Goal: Measurable ⇑h
-- Homeomorph.measurableEmbedding
lemma Homeomorph.measurableEmbedding (h : γ ≃ₜ γ₂) : MeasurableEmbedding h 
Arguments: {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂)
Goal: MeasurableEmbedding ⇑h
-- Homeomorph.toMeasurableEquiv_coe
@[simp]
theorem Homeomorph.toMeasurableEquiv_coe (h : γ ≃ₜ γ₂) : (h.toMeasurableEquiv : γ → γ₂) = h 
Arguments: {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂)
Goal: ⇑h.toMeasurableEquiv = ⇑h
-- Homeomorph.toMeasurableEquiv_symm_coe
@[simp]
theorem Homeomorph.toMeasurableEquiv_symm_coe (h : γ ≃ₜ γ₂) :
    (h.toMeasurableEquiv.symm : γ₂ → γ) = h.symm 
Arguments: {γ : Type u_3} {γ₂ : Type u_4} [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [TopologicalSpace γ₂] [MeasurableSpace γ₂] [BorelSpace γ₂] (h : γ ≃ₜ γ₂)
Goal: ⇑h.toMeasurableEquiv.symm = ⇑h.symm
-- ContinuousMap.measurable
@[measurability]
theorem ContinuousMap.measurable (f : C(α, γ)) : Measurable f 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] (f : C(α, γ))
Goal: Measurable ⇑f
-- measurable_of_continuousOn_compl_singleton
@[fun_prop]
theorem measurable_of_continuousOn_compl_singleton [T1Space α] {f : α → γ} (a : α)
    (hf : ContinuousOn f {a}ᶜ) : Measurable f 
Arguments: {α : Type u_1} {γ : Type u_3} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [T1Space α] {f : α → γ} (a : α) (hf : ContinuousOn f {a}ᶜ)
Goal: Measurable f
-- Continuous.measurable2
theorem Continuous.measurable2 [SecondCountableTopologyEither α β] {f : δ → α}
    {g : δ → β} {c : α → β → γ} (h : Continuous fun p : α × β => c p.1 p.2) (hf : Measurable f)
    (hg : Measurable g) : Measurable fun a => c (f a) (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] [SecondCountableTopologyEither α β] {f : δ → α} {g : δ → β} {c : α → β → γ} (h : Continuous fun p => c p.1 p.2) (hf : Measurable f) (hg : Measurable g)
Goal: Measurable fun a => c (f a) (g a)
-- Continuous.aemeasurable2
theorem Continuous.aemeasurable2 [SecondCountableTopologyEither α β]
    {f : δ → α} {g : δ → β} {c : α → β → γ} {μ : Measure δ}
    (h : Continuous fun p : α × β => c p.1 p.2) (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    AEMeasurable (fun a => c (f a) (g a)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_5} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [MeasurableSpace β] [OpensMeasurableSpace β] [TopologicalSpace γ] [MeasurableSpace γ] [BorelSpace γ] [MeasurableSpace δ] [SecondCountableTopologyEither α β] {f : δ → α} {g : δ → β} {c : α → β → γ} {μ : MeasureTheory.Measure δ} (h : Continuous fun p => c p.1 p.2) (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: AEMeasurable (fun a => c (f a) (g a)) μ
-- pi_le_borel_pi
theorem pi_le_borel_pi {ι : Type*} {π : ι → Type*} [∀ i, TopologicalSpace (π i)]
    [∀ i, MeasurableSpace (π i)] [∀ i, BorelSpace (π i)] :
      MeasurableSpace.pi ≤ borel (∀ i, π i) 
Arguments: {ι : Type u_6} {π : ι → Type u_7} [(i : ι) → TopologicalSpace (π i)] [(i : ι) → MeasurableSpace (π i)] [∀ (i : ι), BorelSpace (π i)]
Goal: MeasurableSpace.pi ≤ borel ((i : ι) → π i)
-- prod_le_borel_prod
theorem prod_le_borel_prod : Prod.instMeasurableSpace ≤ borel (α × β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [TopologicalSpace β] [MeasurableSpace β] [BorelSpace β]
Goal: Prod.instMeasurableSpace ≤ borel (α × β)
-- MeasurableEmbedding.borelSpace
lemma MeasurableEmbedding.borelSpace {α β : Type*} [MeasurableSpace α] [TopologicalSpace α]
    [MeasurableSpace β] [TopologicalSpace β] [hβ : BorelSpace β] {e : α → β}
    (h'e : MeasurableEmbedding e) (h''e : Inducing e) :
    BorelSpace α 
Arguments: {α : Type u_6} {β : Type u_7} [MeasurableSpace α] [TopologicalSpace α] [MeasurableSpace β] [TopologicalSpace β] [hβ : BorelSpace β] {e : α → β} (h'e : MeasurableEmbedding e) (h''e : Inducing e)
Goal: BorelSpace α