ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ContinuousOn.integrableAt_nhdsWithin
theorem ContinuousOn.integrableAt_nhdsWithin [TopologicalSpace α]
    [SecondCountableTopologyEither α E] [OpensMeasurableSpace α] {μ : Measure α}
    [IsLocallyFiniteMeasure μ] {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t)
    (ht : MeasurableSet t) (ha : a ∈ t) : IntegrableAtFilter f (𝓝[t] a) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [TopologicalSpace α] [SecondCountableTopologyEither α E] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {a : α} {t : Set α} {f : α → E} (hft : ContinuousOn f t) (ht : MeasurableSet t) (ha : a ∈ t)
Goal: MeasureTheory.IntegrableAtFilter f (𝓝[t] a) μ
-- Continuous.integrableAt_nhds
theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
    [OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
    (hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [TopologicalSpace α] [SecondCountableTopologyEither α E] [OpensMeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsLocallyFiniteMeasure μ] {f : α → E} (hf : Continuous f) (a : α)
Goal: MeasureTheory.IntegrableAtFilter f (𝓝 a) μ
-- ContinuousOn.stronglyMeasurableAtFilter
theorem ContinuousOn.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β}
    {s : Set α} {μ : Measure α} (hs : IsOpen s) (hf : ContinuousOn f s) :
    ∀ x ∈ s, StronglyMeasurableAtFilter f (𝓝 x) μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hs : IsOpen s) (hf : ContinuousOn f s) (x : α)
Goal: x ∈ s → StronglyMeasurableAtFilter f (𝓝 x) μ
-- ContinuousAt.stronglyMeasurableAtFilter
theorem ContinuousAt.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [SecondCountableTopologyEither α E] {f : α → E} {s : Set α} {μ : Measure α} (hs : IsOpen s)
    (hf : ∀ x ∈ s, ContinuousAt f x) : ∀ x ∈ s, StronglyMeasurableAtFilter f (𝓝 x) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [TopologicalSpace α] [OpensMeasurableSpace α] [SecondCountableTopologyEither α E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hs : IsOpen s) (hf : ∀ x ∈ s, ContinuousAt f x) (x : α)
Goal: x ∈ s → StronglyMeasurableAtFilter f (𝓝 x) μ
-- Continuous.stronglyMeasurableAtFilter
theorem Continuous.stronglyMeasurableAtFilter [TopologicalSpace α] [OpensMeasurableSpace α]
    [TopologicalSpace β] [PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β}
    (hf : Continuous f) (μ : Measure α) (l : Filter α) : StronglyMeasurableAtFilter f l μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β} (hf : Continuous f) (μ : MeasureTheory.Measure α) (l : Filter α)
Goal: StronglyMeasurableAtFilter f l μ
-- ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin
theorem ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin {α β : Type*} [MeasurableSpace α]
    [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [PseudoMetrizableSpace β]
    [SecondCountableTopologyEither α β] {f : α → β} {s : Set α} {μ : Measure α}
    (hf : ContinuousOn f s) (hs : MeasurableSet s) (x : α) :
    StronglyMeasurableAtFilter f (𝓝[s] x) μ 
Arguments: {α : Type u_5} {β : Type u_6} [MeasurableSpace α] [TopologicalSpace α] [OpensMeasurableSpace α] [TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopologyEither α β] {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α} (hf : ContinuousOn f s) (hs : MeasurableSet s) (x : α)
Goal: StronglyMeasurableAtFilter f (𝓝[s] x) μ
-- integrableOn_Icc_iff_integrableOn_Ioc'
theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} (ha : μ {a} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioc a b) μ
-- integrableOn_Icc_iff_integrableOn_Ico'
theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} (hb : μ {b} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ico a b) μ
-- integrableOn_Ico_iff_integrableOn_Ioo'
theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} (ha : μ {a} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Ico a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Ioc_iff_integrableOn_Ioo'
theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} (hb : μ {b} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Icc_iff_integrableOn_Ioo'
theorem integrableOn_Icc_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} (ha : μ {a} ≠ ⊤) (hb : μ {b} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Ici_iff_integrableOn_Ioi'
theorem integrableOn_Ici_iff_integrableOn_Ioi' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {b : α} (hb : μ {b} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Ici b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioi b) μ
-- integrableOn_Iic_iff_integrableOn_Iio'
theorem integrableOn_Iic_iff_integrableOn_Iio' (hb : μ {b} ≠ ∞) :
    IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {b : α} (hb : μ {b} ≠ ⊤)
Goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ ↔ MeasureTheory.IntegrableOn f (Set.Iio b) μ
-- integrableOn_Icc_iff_integrableOn_Ioc
theorem integrableOn_Icc_iff_integrableOn_Ioc :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioc a b) μ
-- integrableOn_Icc_iff_integrableOn_Ico
theorem integrableOn_Icc_iff_integrableOn_Ico :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ico a b) μ
-- integrableOn_Ico_iff_integrableOn_Ioo
theorem integrableOn_Ico_iff_integrableOn_Ioo :
    IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Ico a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Ioc_iff_integrableOn_Ioo
theorem integrableOn_Ioc_iff_integrableOn_Ioo :
    IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Ioc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Icc_iff_integrableOn_Ioo
theorem integrableOn_Icc_iff_integrableOn_Ioo :
    IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {a b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Icc a b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- integrableOn_Ici_iff_integrableOn_Ioi
theorem integrableOn_Ici_iff_integrableOn_Ioi :
    IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Ici b) μ ↔ MeasureTheory.IntegrableOn f (Set.Ioi b) μ
-- integrableOn_Iic_iff_integrableOn_Iio
theorem integrableOn_Iic_iff_integrableOn_Iio :
    IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] [PartialOrder α] [MeasurableSingletonClass α] {f : α → E} {μ : MeasureTheory.Measure α} {b : α} [MeasureTheory.NoAtoms μ]
Goal: MeasureTheory.IntegrableOn f (Set.Iic b) μ ↔ MeasureTheory.IntegrableOn f (Set.Iio b) μ
-- Complex.circleTransformDeriv_periodic
theorem circleTransformDeriv_periodic (f : ℂ → E) :
    Periodic (circleTransformDeriv R z w f) (2 * π) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (R : ℝ) (z w : ℂ) (f : ℂ → E)
Goal: Function.Periodic (Complex.circleTransformDeriv R z w f) (2 * π)
-- Complex.circleTransformDeriv_eq
theorem circleTransformDeriv_eq (f : ℂ → E) : circleTransformDeriv R z w f =
    fun θ => (circleMap z R θ - w)⁻¹ • circleTransform R z w f θ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (R : ℝ) (z w : ℂ) (f : ℂ → E)
Goal: Complex.circleTransformDeriv R z w f = fun θ => (circleMap z R θ - w)⁻¹ • Complex.circleTransform R z w f θ
-- Complex.integral_circleTransform
theorem integral_circleTransform (f : ℂ → E) :
    (∫ θ : ℝ in (0)..2 * π, circleTransform R z w f θ) =
      (2 * ↑π * I)⁻¹ • ∮ z in C(z, R), (z - w)⁻¹ • f z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (R : ℝ) (z w : ℂ) (f : ℂ → E)
Goal: ∫ (θ : ℝ) in 0 ..2 * π, Complex.circleTransform R z w f θ =
  (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z
-- Complex.continuous_circleTransform
theorem continuous_circleTransform {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
    (hf : ContinuousOn f <| sphere z R) (hw : w ∈ ball z R) :
    Continuous (circleTransform R z w f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ} (hf : ContinuousOn f (Metric.sphere z R)) (hw : w ∈ Metric.ball z R)
Goal: Continuous (Complex.circleTransform R z w f)
-- Complex.continuous_circleTransformDeriv
theorem continuous_circleTransformDeriv {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
    (hf : ContinuousOn f (sphere z R)) (hw : w ∈ ball z R) :
    Continuous (circleTransformDeriv R z w f) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ} (hf : ContinuousOn f (Metric.sphere z R)) (hw : w ∈ Metric.ball z R)
Goal: Continuous (Complex.circleTransformDeriv R z w f)
-- Complex.continuousOn_prod_circle_transform_function
theorem continuousOn_prod_circle_transform_function {R r : ℝ} (hr : r < R) {z : ℂ} :
    ContinuousOn (fun w : ℂ × ℝ => (circleMap z R w.snd - w.fst)⁻¹ ^ 2)
      (closedBall z r ×ˢ univ) 
Arguments: {R r : ℝ} (hr : r < R) {z : ℂ}
Goal: ContinuousOn (fun w => (circleMap z R w.2 - w.1)⁻¹ ^ 2) (Metric.closedBall z r ×ˢ Set.univ)
-- Complex.continuousOn_abs_circleTransformBoundingFunction
theorem continuousOn_abs_circleTransformBoundingFunction {R r : ℝ} (hr : r < R) (z : ℂ) :
    ContinuousOn (abs ∘ circleTransformBoundingFunction R z) (closedBall z r ×ˢ univ) 
Arguments: {R r : ℝ} (hr : r < R) (z : ℂ)
Goal: ContinuousOn (⇑Complex.abs ∘ Complex.circleTransformBoundingFunction R z) (Metric.closedBall z r ×ˢ Set.univ)
-- Complex.abs_circleTransformBoundingFunction_le
theorem abs_circleTransformBoundingFunction_le {R r : ℝ} (hr : r < R) (hr' : 0 ≤ r) (z : ℂ) :
    ∃ x : closedBall z r ×ˢ [[0, 2 * π]], ∀ y : closedBall z r ×ˢ [[0, 2 * π]],
    abs (circleTransformBoundingFunction R z y) ≤ abs (circleTransformBoundingFunction R z x) 
Arguments: {R r : ℝ} (hr : r < R) (hr' : 0 ≤ r) (z : ℂ)
Goal: ∃ x,
  ∀ (y : ↑(Metric.closedBall z r ×ˢ [[0, 2 * π]])),
    Complex.abs (Complex.circleTransformBoundingFunction R z ↑y) ≤
      Complex.abs (Complex.circleTransformBoundingFunction R z ↑x)
-- Complex.circleTransformDeriv_bound
theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R)
    (hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧
      ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B 
Arguments: {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ Metric.ball z R) (hf : ContinuousOn f (Metric.sphere z R))
Goal: ∃ B ε,
  0 < ε ∧
    Metric.ball x ε ⊆ Metric.ball z R ∧ ∀ (t : ℝ), ∀ y ∈ Metric.ball x ε, ‖Complex.circleTransformDeriv R z y f t‖ ≤ B
-- MeasureTheory.IsSetSemiring.inter_mem
structure IsSetSemiring (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  inter_mem : ∀ s ∈ C, ∀ t ∈ C, s ∩ t ∈ C
  diff_eq_sUnion' : ∀ s ∈ C, ∀ t ∈ C,
    ∃ I : Finset (Set α), ↑I ⊆ C ∧ PairwiseDisjoint (I : Set (Set α)) id ∧ s \ t = ⋃₀ I
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetSemiring C) (s : Set α)
Goal: s ∈ C → ∀ t ∈ C, s ∩ t ∈ C
-- MeasureTheory.IsSetSemiring.empty_mem
structure IsSetSemiring (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  inter_mem : ∀ s ∈ C, ∀ t ∈ C, s ∩ t ∈ C
  diff_eq_sUnion' : ∀ s ∈ C, ∀ t ∈ C,
    ∃ I : Finset (Set α), ↑I ⊆ C ∧ PairwiseDisjoint (I : Set (Set α)) id ∧ s \ t = ⋃₀ I
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetSemiring C)
Goal: ∅ ∈ C
-- MeasureTheory.IsSetSemiring.diff_eq_sUnion'
structure IsSetSemiring (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  inter_mem : ∀ s ∈ C, ∀ t ∈ C, s ∩ t ∈ C
  diff_eq_sUnion' : ∀ s ∈ C, ∀ t ∈ C,
    ∃ I : Finset (Set α), ↑I ⊆ C ∧ PairwiseDisjoint (I : Set (Set α)) id ∧ s \ t = ⋃₀ I
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetSemiring C) (s : Set α)
Goal: s ∈ C → ∀ t ∈ C, ∃ I, ↑I ⊆ C ∧ (↑I).PairwiseDisjoint id ∧ s \ t = ⋃₀ ↑I
-- MeasureTheory.IsSetSemiring.isPiSystem
lemma isPiSystem (hC : IsSetSemiring C) : IsPiSystem C 
Arguments: {α : Type u_1} {C : Set (Set α)} (hC : MeasureTheory.IsSetSemiring C)
Goal: IsPiSystem C
-- MeasureTheory.IsSetSemiring.empty_not_mem_diffFinset
lemma empty_not_mem_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    ∅ ∉ hC.diffFinset hs ht 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: ∅ ∉ hC.diffFinset hs ht
-- MeasureTheory.IsSetSemiring.diffFinset_subset
lemma diffFinset_subset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    ↑(hC.diffFinset hs ht) ⊆ C 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: ↑(hC.diffFinset hs ht) ⊆ C
-- MeasureTheory.IsSetSemiring.pairwiseDisjoint_diffFinset
lemma pairwiseDisjoint_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    PairwiseDisjoint (hC.diffFinset hs ht : Set (Set α)) id 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: (↑(hC.diffFinset hs ht)).PairwiseDisjoint id
-- MeasureTheory.IsSetSemiring.sUnion_diffFinset
lemma sUnion_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    ⋃₀ hC.diffFinset hs ht = s \ t 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: ⋃₀ ↑(hC.diffFinset hs ht) = s \ t
-- MeasureTheory.IsSetSemiring.not_mem_diffFinset
lemma not_mem_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    t ∉ hC.diffFinset hs ht 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: t ∉ hC.diffFinset hs ht
-- MeasureTheory.IsSetSemiring.sUnion_insert_diffFinset
lemma sUnion_insert_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) (hst : t ⊆ s) :
    ⋃₀ insert t (hC.diffFinset hs ht) = s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) (hst : t ⊆ s)
Goal: ⋃₀ insert t ↑(hC.diffFinset hs ht) = s
-- MeasureTheory.IsSetSemiring.disjoint_sUnion_diffFinset
lemma disjoint_sUnion_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    Disjoint t (⋃₀ hC.diffFinset hs ht) 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: Disjoint t (⋃₀ ↑(hC.diffFinset hs ht))
-- MeasureTheory.IsSetSemiring.pairwiseDisjoint_insert_diffFinset
lemma pairwiseDisjoint_insert_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
    PairwiseDisjoint (insert t (hC.diffFinset hs ht) : Set (Set α)) id 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
Goal: (insert t ↑(hC.diffFinset hs ht)).PairwiseDisjoint id
-- MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq
lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    ∃ J : Finset (Set α), ↑J ⊆ C ∧ PairwiseDisjoint (J : Set (Set α)) id ∧
      s \ ⋃₀ I = ⋃₀ J 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \ ⋃₀ ↑I = ⋃₀ ↑J
-- MeasureTheory.IsSetSemiring.empty_not_mem_diffFinset₀
lemma empty_not_mem_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    ∅ ∉ hC.diffFinset₀ hs hI 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: ∅ ∉ hC.diffFinset₀ hs hI
-- MeasureTheory.IsSetSemiring.diffFinset₀_subset
lemma diffFinset₀_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    ↑(hC.diffFinset₀ hs hI) ⊆ C 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: ↑(hC.diffFinset₀ hs hI) ⊆ C
-- MeasureTheory.IsSetSemiring.pairwiseDisjoint_diffFinset₀
lemma pairwiseDisjoint_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    PairwiseDisjoint (hC.diffFinset₀ hs hI : Set (Set α)) id 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: (↑(hC.diffFinset₀ hs hI)).PairwiseDisjoint id
-- MeasureTheory.IsSetSemiring.diff_sUnion_eq_sUnion_diffFinset₀
lemma diff_sUnion_eq_sUnion_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    s \ ⋃₀ I = ⋃₀ hC.diffFinset₀ hs hI 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: s \ ⋃₀ ↑I = ⋃₀ ↑(hC.diffFinset₀ hs hI)
-- MeasureTheory.IsSetSemiring.sUnion_diffFinset₀_subset
lemma sUnion_diffFinset₀_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    ⋃₀ (hC.diffFinset₀ hs hI : Set (Set α)) ⊆ s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: ⋃₀ ↑(hC.diffFinset₀ hs hI) ⊆ s
-- MeasureTheory.IsSetSemiring.disjoint_sUnion_diffFinset₀
lemma disjoint_sUnion_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    Disjoint (⋃₀ (I : Set (Set α))) (⋃₀ hC.diffFinset₀ hs hI) 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: Disjoint (⋃₀ ↑I) (⋃₀ ↑(hC.diffFinset₀ hs hI))
-- MeasureTheory.IsSetSemiring.disjoint_diffFinset₀
lemma disjoint_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
    Disjoint I (hC.diffFinset₀ hs hI) 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
Goal: Disjoint I (hC.diffFinset₀ hs hI)
-- MeasureTheory.IsSetSemiring.pairwiseDisjoint_union_diffFinset₀
lemma pairwiseDisjoint_union_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C)
    (hI : ↑I ⊆ C) (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
    PairwiseDisjoint (I ∪ hC.diffFinset₀ hs hI : Set (Set α)) id 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) (h_dis : (↑I).PairwiseDisjoint id)
Goal: (↑I ∪ ↑(hC.diffFinset₀ hs hI)).PairwiseDisjoint id
-- MeasureTheory.IsSetSemiring.sUnion_union_sUnion_diffFinset₀_of_subset
lemma sUnion_union_sUnion_diffFinset₀_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
    (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) :
    ⋃₀ I ∪ ⋃₀ hC.diffFinset₀ hs hI = s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s)
Goal: ⋃₀ ↑I ∪ ⋃₀ ↑(hC.diffFinset₀ hs hI) = s
-- MeasureTheory.IsSetSemiring.sUnion_union_diffFinset₀_of_subset
lemma sUnion_union_diffFinset₀_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
    (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) [DecidableEq (Set α)] :
    ⋃₀ ↑(I ∪ hC.diffFinset₀ hs hI) = s 
Arguments: {α : Type u_1} {C : Set (Set α)} {s : Set α} {I : Finset (Set α)} (hC : MeasureTheory.IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) [DecidableEq (Set α)]
Goal: ⋃₀ ↑(I ∪ hC.diffFinset₀ hs hI) = s
-- MeasureTheory.IsSetRing.union_mem
structure IsSetRing (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  union_mem ⦃s t⦄ : s ∈ C → t ∈ C → s ∪ t ∈ C
  diff_mem ⦃s t⦄ : s ∈ C → t ∈ C → s \ t ∈ C
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetRing C) ⦃s t : Set α⦄
Goal: s ∈ C → t ∈ C → s ∪ t ∈ C
-- MeasureTheory.IsSetRing.diff_mem
structure IsSetRing (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  union_mem ⦃s t⦄ : s ∈ C → t ∈ C → s ∪ t ∈ C
  diff_mem ⦃s t⦄ : s ∈ C → t ∈ C → s \ t ∈ C
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetRing C) ⦃s t : Set α⦄
Goal: s ∈ C → t ∈ C → s \ t ∈ C
-- MeasureTheory.IsSetRing.empty_mem
structure IsSetRing (C : Set (Set α)) : Prop where
  empty_mem : ∅ ∈ C
  union_mem ⦃s t⦄ : s ∈ C → t ∈ C → s ∪ t ∈ C
  diff_mem ⦃s t⦄ : s ∈ C → t ∈ C → s \ t ∈ C
Arguments: {α : Type u_1} {C : Set (Set α)} (self : MeasureTheory.IsSetRing C)
Goal: ∅ ∈ C
-- MeasureTheory.IsSetRing.inter_mem
lemma inter_mem (hC : IsSetRing C) (hs : s ∈ C) (ht : t ∈ C) : s ∩ t ∈ C 
Arguments: {α : Type u_1} {C : Set (Set α)} {s t : Set α} (hC : MeasureTheory.IsSetRing C) (hs : s ∈ C) (ht : t ∈ C)
Goal: s ∩ t ∈ C