ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AEMeasurable.coe_nnreal_ennreal
@[measurability, fun_prop]
theorem AEMeasurable.coe_nnreal_ennreal {f : α → ℝ≥0} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x : ℝ≥0∞)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ↑(f x)) μ
-- Measurable.ennreal_ofReal
@[measurability, fun_prop]
theorem Measurable.ennreal_ofReal {f : α → ℝ} (hf : Measurable f) :
    Measurable fun x => ENNReal.ofReal (f x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => ENNReal.ofReal (f x)
-- AEMeasurable.ennreal_ofReal
@[measurability, fun_prop]
lemma AEMeasurable.ennreal_ofReal {f : α → ℝ} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x ↦ ENNReal.ofReal (f x)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ENNReal.ofReal (f x)) μ
-- measurable_coe_nnreal_real_iff
@[simp, norm_cast]
theorem measurable_coe_nnreal_real_iff {f : α → ℝ≥0} :
    Measurable (fun x => f x : α → ℝ) ↔ Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0}
Goal: (Measurable fun x => ↑(f x)) ↔ Measurable f
-- aemeasurable_coe_nnreal_real_iff
@[simp, norm_cast]
theorem aemeasurable_coe_nnreal_real_iff {f : α → ℝ≥0} {μ : Measure α} :
    AEMeasurable (fun x => f x : α → ℝ) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} {μ : MeasureTheory.Measure α}
Goal: AEMeasurable (fun x => ↑(f x)) μ ↔ AEMeasurable f μ
-- ENNReal.measurable_of_measurable_nnreal
theorem measurable_of_measurable_nnreal {f : ℝ≥0∞ → α} (h : Measurable fun p : ℝ≥0 => f p) :
    Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : ℝ≥0∞ → α} (h : Measurable fun p => f ↑p)
Goal: Measurable f
-- ENNReal.measurable_of_measurable_nnreal_prod
theorem measurable_of_measurable_nnreal_prod [MeasurableSpace β] [MeasurableSpace γ]
    {f : ℝ≥0∞ × β → γ} (H₁ : Measurable fun p : ℝ≥0 × β => f (p.1, p.2))
    (H₂ : Measurable fun x => f (∞, x)) : Measurable f 
Arguments: {β : Type u_2} {γ : Type u_3} [MeasurableSpace β] [MeasurableSpace γ] {f : ℝ≥0∞ × β → γ} (H₁ : Measurable fun p => f (↑p.1, p.2)) (H₂ : Measurable fun x => f (⊤, x))
Goal: Measurable f
-- ENNReal.measurable_of_measurable_nnreal_nnreal
theorem measurable_of_measurable_nnreal_nnreal [MeasurableSpace β] {f : ℝ≥0∞ × ℝ≥0∞ → β}
    (h₁ : Measurable fun p : ℝ≥0 × ℝ≥0 => f (p.1, p.2)) (h₂ : Measurable fun r : ℝ≥0 => f (∞, r))
    (h₃ : Measurable fun r : ℝ≥0 => f (r, ∞)) : Measurable f 
Arguments: {β : Type u_2} [MeasurableSpace β] {f : ℝ≥0∞ × ℝ≥0∞ → β} (h₁ : Measurable fun p => f (↑p.1, ↑p.2)) (h₂ : Measurable fun r => f (⊤, ↑r)) (h₃ : Measurable fun r => f (↑r, ⊤))
Goal: Measurable f
-- ENNReal.measurable_ofReal
@[measurability]
theorem measurable_ofReal : Measurable ENNReal.ofReal 
Goal: Measurable ENNReal.ofReal
-- ENNReal.measurable_toReal
@[measurability]
theorem measurable_toReal : Measurable ENNReal.toReal 
Goal: Measurable ENNReal.toReal
-- ENNReal.measurable_toNNReal
@[measurability]
theorem measurable_toNNReal : Measurable ENNReal.toNNReal 
Goal: Measurable ENNReal.toNNReal
-- ENNReal.measurable_of_tendsto'
theorem measurable_of_tendsto' {ι : Type*} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞} (u : Filter ι)
    [NeBot u] [IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
    Measurable g 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞} (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] (hf : ∀ (i : ι), Measurable (f i)) (lim : Filter.Tendsto f u (𝓝 g))
Goal: Measurable g
-- ENNReal.measurable_of_tendsto
theorem measurable_of_tendsto {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} (hf : ∀ i, Measurable (f i))
    (lim : Tendsto f atTop (𝓝 g)) : Measurable g 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} (hf : ∀ (i : ℕ), Measurable (f i)) (lim : Filter.Tendsto f Filter.atTop (𝓝 g))
Goal: Measurable g
-- ENNReal.aemeasurable_of_tendsto'
lemma aemeasurable_of_tendsto' {ι : Type*} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞}
    {μ : Measure α} (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
    (hf : ∀ i, AEMeasurable (f i) μ) (hlim : ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))) :
    AEMeasurable g μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} {f : ι → α → ℝ≥0∞} {g : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] (hf : ∀ (i : ι), AEMeasurable (f i) μ) (hlim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) u (𝓝 (g a)))
Goal: AEMeasurable g μ
-- ENNReal.aemeasurable_of_tendsto
lemma aemeasurable_of_tendsto {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} {μ : Measure α}
    (hf : ∀ i, AEMeasurable (f i) μ) (hlim : ∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (g a))) :
    AEMeasurable g μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : ℕ → α → ℝ≥0∞} {g : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf : ∀ (i : ℕ), AEMeasurable (f i) μ) (hlim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun i => f i a) Filter.atTop (𝓝 (g a)))
Goal: AEMeasurable g μ
-- Measurable.ennreal_toNNReal
@[measurability, fun_prop]
theorem Measurable.ennreal_toNNReal {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x => (f x).toNNReal 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable fun x => (f x).toNNReal
-- AEMeasurable.ennreal_toNNReal
@[measurability, fun_prop]
theorem AEMeasurable.ennreal_toNNReal {f : α → ℝ≥0∞} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x).toNNReal) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => (f x).toNNReal) μ
-- measurable_coe_nnreal_ennreal_iff
@[simp, norm_cast]
theorem measurable_coe_nnreal_ennreal_iff {f : α → ℝ≥0} :
    (Measurable fun x => (f x : ℝ≥0∞)) ↔ Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0}
Goal: (Measurable fun x => ↑(f x)) ↔ Measurable f
-- aemeasurable_coe_nnreal_ennreal_iff
@[simp, norm_cast]
theorem aemeasurable_coe_nnreal_ennreal_iff {f : α → ℝ≥0} {μ : Measure α} :
    AEMeasurable (fun x => (f x : ℝ≥0∞)) μ ↔ AEMeasurable f μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0} {μ : MeasureTheory.Measure α}
Goal: AEMeasurable (fun x => ↑(f x)) μ ↔ AEMeasurable f μ
-- Measurable.ennreal_toReal
@[measurability, fun_prop]
theorem Measurable.ennreal_toReal {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x => ENNReal.toReal (f x) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable fun x => (f x).toReal
-- AEMeasurable.ennreal_toReal
@[measurability, fun_prop]
theorem AEMeasurable.ennreal_toReal {f : α → ℝ≥0∞} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => ENNReal.toReal (f x)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => (f x).toReal) μ
-- Measurable.ennreal_tsum
@[measurability, fun_prop]
theorem Measurable.ennreal_tsum {ι} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
    Measurable fun x => ∑' i, f i x 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), Measurable (f i))
Goal: Measurable fun x => ∑' (i : ι), f i x
-- Measurable.ennreal_tsum'
@[measurability, fun_prop]
theorem Measurable.ennreal_tsum' {ι} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ i, Measurable (f i)) :
    Measurable (∑' i, f i) 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} [Countable ι] {f : ι → α → ℝ≥0∞} (h : ∀ (i : ι), Measurable (f i))
Goal: Measurable (∑' (i : ι), f i)
-- Measurable.nnreal_tsum
@[measurability, fun_prop]
theorem Measurable.nnreal_tsum {ι} [Countable ι] {f : ι → α → ℝ≥0} (h : ∀ i, Measurable (f i)) :
    Measurable fun x => ∑' i, f i x 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} [Countable ι] {f : ι → α → ℝ≥0} (h : ∀ (i : ι), Measurable (f i))
Goal: Measurable fun x => ∑' (i : ι), f i x
-- AEMeasurable.ennreal_tsum
@[measurability, fun_prop]
theorem AEMeasurable.ennreal_tsum {ι} [Countable ι] {f : ι → α → ℝ≥0∞} {μ : Measure α}
    (h : ∀ i, AEMeasurable (f i) μ) : AEMeasurable (fun x => ∑' i, f i x) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} [Countable ι] {f : ι → α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (h : ∀ (i : ι), AEMeasurable (f i) μ)
Goal: AEMeasurable (fun x => ∑' (i : ι), f i x) μ
-- AEMeasurable.nnreal_tsum
@[measurability, fun_prop]
theorem AEMeasurable.nnreal_tsum {α : Type*} [MeasurableSpace α] {ι : Type*} [Countable ι]
    {f : ι → α → NNReal} {μ : MeasureTheory.Measure α} (h : ∀ i : ι, AEMeasurable (f i) μ) :
    AEMeasurable (fun x : α => ∑' i : ι, f i x) μ 
Arguments: {α : Type u_5} [MeasurableSpace α] {ι : Type u_6} [Countable ι] {f : ι → α → ℝ≥0} {μ : MeasureTheory.Measure α} (h : ∀ (i : ι), AEMeasurable (f i) μ)
Goal: AEMeasurable (fun x => ∑' (i : ι), f i x) μ
-- measurable_coe_real_ereal
@[measurability, fun_prop]
theorem measurable_coe_real_ereal : Measurable ((↑) : ℝ → EReal) 
Goal: Measurable Real.toEReal
-- Measurable.coe_real_ereal
@[measurability]
theorem Measurable.coe_real_ereal {f : α → ℝ} (hf : Measurable f) :
    Measurable fun x => (f x : EReal) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (hf : Measurable f)
Goal: Measurable fun x => ↑(f x)
-- AEMeasurable.coe_real_ereal
@[measurability]
theorem AEMeasurable.coe_real_ereal {f : α → ℝ} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x : EReal)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ↑(f x)) μ
-- EReal.measurable_of_measurable_real
theorem EReal.measurable_of_measurable_real {f : EReal → α} (h : Measurable fun p : ℝ => f p) :
    Measurable f 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : EReal → α} (h : Measurable fun p => f ↑p)
Goal: Measurable f
-- measurable_ereal_toReal
@[measurability]
theorem measurable_ereal_toReal : Measurable EReal.toReal 
Goal: Measurable EReal.toReal
-- Measurable.ereal_toReal
@[measurability, fun_prop]
theorem Measurable.ereal_toReal {f : α → EReal} (hf : Measurable f) :
    Measurable fun x => (f x).toReal 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → EReal} (hf : Measurable f)
Goal: Measurable fun x => (f x).toReal
-- AEMeasurable.ereal_toReal
@[measurability, fun_prop]
theorem AEMeasurable.ereal_toReal {f : α → EReal} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x).toReal) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → EReal} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => (f x).toReal) μ
-- measurable_coe_ennreal_ereal
@[measurability]
theorem measurable_coe_ennreal_ereal : Measurable ((↑) : ℝ≥0∞ → EReal) 
Goal: Measurable ENNReal.toEReal
-- Measurable.coe_ereal_ennreal
@[measurability, fun_prop]
theorem Measurable.coe_ereal_ennreal {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x => (f x : EReal) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} (hf : Measurable f)
Goal: Measurable fun x => ↑(f x)
-- AEMeasurable.coe_ereal_ennreal
@[measurability, fun_prop]
theorem AEMeasurable.coe_ereal_ennreal {f : α → ℝ≥0∞} {μ : Measure α} (hf : AEMeasurable f μ) :
    AEMeasurable (fun x => (f x : EReal)) μ 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ≥0∞} {μ : MeasureTheory.Measure α} (hf : AEMeasurable f μ)
Goal: AEMeasurable (fun x => ↑(f x)) μ
-- NNReal.measurable_of_tendsto'
theorem measurable_of_tendsto' {ι} {f : ι → α → ℝ≥0} {g : α → ℝ≥0} (u : Filter ι) [NeBot u]
    [IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
    Measurable g 
Arguments: {α : Type u_1} [MeasurableSpace α] {ι : Type u_5} {f : ι → α → ℝ≥0} {g : α → ℝ≥0} (u : Filter ι) [u.NeBot] [u.IsCountablyGenerated] (hf : ∀ (i : ι), Measurable (f i)) (lim : Filter.Tendsto f u (𝓝 g))
Goal: Measurable g
-- NNReal.measurable_of_tendsto
theorem measurable_of_tendsto {f : ℕ → α → ℝ≥0} {g : α → ℝ≥0} (hf : ∀ i, Measurable (f i))
    (lim : Tendsto f atTop (𝓝 g)) : Measurable g 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : ℕ → α → ℝ≥0} {g : α → ℝ≥0} (hf : ∀ (i : ℕ), Measurable (f i)) (lim : Filter.Tendsto f Filter.atTop (𝓝 g))
Goal: Measurable g
-- exists_spanning_measurableSet_le
theorem exists_spanning_measurableSet_le {m : MeasurableSpace α} {f : α → ℝ≥0}
    (hf : Measurable f) (μ : Measure α) [SigmaFinite μ] :
    ∃ s : ℕ → Set α,
      (∀ n, MeasurableSet (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, f x ≤ n) ∧
      ⋃ i, s i = Set.univ 
Arguments: {α : Type u_1} {m : MeasurableSpace α} {f : α → ℝ≥0} (hf : Measurable f) (μ : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ]
Goal: ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ⊤ ∧ ∀ x ∈ s n, f x ≤ ↑n) ∧ ⋃ i, s i = Set.univ
-- tendsto_measure_Icc_nhdsWithin_right'
lemma tendsto_measure_Icc_nhdsWithin_right' (b : ℝ) :
    Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[>] (0 : ℝ)) (𝓝 (μ {b})) 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasureOnCompacts μ] (b : ℝ)
Goal: Filter.Tendsto (fun δ => μ (Set.Icc (b - δ) (b + δ))) (𝓝[>] 0) (𝓝 (μ {b}))
-- tendsto_measure_Icc_nhdsWithin_right
lemma tendsto_measure_Icc_nhdsWithin_right (b : ℝ) :
    Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[≥] (0 : ℝ)) (𝓝 (μ {b})) 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasureOnCompacts μ] (b : ℝ)
Goal: Filter.Tendsto (fun δ => μ (Set.Icc (b - δ) (b + δ))) (𝓝[≥] 0) (𝓝 (μ {b}))
-- tendsto_measure_Icc
lemma tendsto_measure_Icc [NoAtoms μ] (b : ℝ) :
    Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝 (0 : ℝ)) (𝓝 0) 
Arguments: (μ : MeasureTheory.Measure ℝ) [MeasureTheory.IsFiniteMeasureOnCompacts μ] [MeasureTheory.NoAtoms μ] (b : ℝ)
Goal: Filter.Tendsto (fun δ => μ (Set.Icc (b - δ) (b + δ))) (𝓝 0) (𝓝 0)
-- MeasureTheory.squareCylinders_eq_iUnion_image
theorem squareCylinders_eq_iUnion_image (C : ∀ i, Set (Set (α i))) :
    squareCylinders C = ⋃ s : Finset ι, (fun t ↦ (s : Set ι).pi t) '' univ.pi C 
Arguments: {ι : Type u_2} {α : ι → Type u_1} (C : (i : ι) → Set (Set (α i)))
Goal: MeasureTheory.squareCylinders C = ⋃ s, (fun t => (↑s).pi t) '' Set.univ.pi C
-- MeasureTheory.isPiSystem_squareCylinders
theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
    (hC_univ : ∀ i, univ ∈ C i) :
    IsPiSystem (squareCylinders C) 
Arguments: {ι : Type u_2} {α : ι → Type u_1} {C : (i : ι) → Set (Set (α i))} (hC : ∀ (i : ι), IsPiSystem (C i)) (hC_univ : ∀ (i : ι), Set.univ ∈ C i)
Goal: IsPiSystem (MeasureTheory.squareCylinders C)
-- MeasureTheory.comap_eval_le_generateFrom_squareCylinders_singleton
theorem comap_eval_le_generateFrom_squareCylinders_singleton
    (α : ι → Type*) [m : ∀ i, MeasurableSpace (α i)] (i : ι) :
    MeasurableSpace.comap (Function.eval i) (m i) ≤
      MeasurableSpace.generateFrom
        ((fun t ↦ ({i} : Set ι).pi t) '' univ.pi fun i ↦ {s : Set (α i) | MeasurableSet s}) 
Arguments: {ι : Type u_2} (α : ι → Type u_1) [m : (i : ι) → MeasurableSpace (α i)] (i : ι)
Goal: MeasurableSpace.comap (Function.eval i) (m i) ≤
  MeasurableSpace.generateFrom ((fun t => {i}.pi t) '' Set.univ.pi fun i => {s | MeasurableSet s})
-- MeasureTheory.generateFrom_squareCylinders
theorem generateFrom_squareCylinders [∀ i, MeasurableSpace (α i)] :
    MeasurableSpace.generateFrom (squareCylinders fun i ↦ {s : Set (α i) | MeasurableSet s}) =
      MeasurableSpace.pi 
Arguments: {ι : Type u_2} {α : ι → Type u_1} [(i : ι) → MeasurableSpace (α i)]
Goal: MeasurableSpace.generateFrom (MeasureTheory.squareCylinders fun i => {s | MeasurableSet s}) = MeasurableSpace.pi
-- MeasureTheory.mem_cylinder
@[simp]
theorem mem_cylinder (s : Finset ι) (S : Set (∀ i : s, α i)) (f : ∀ i, α i) :
    f ∈ cylinder s S ↔ (fun i : s ↦ f i) ∈ S 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι) (S : Set ((i : { x // x ∈ s }) → α ↑i)) (f : (i : ι) → α i)
Goal: f ∈ MeasureTheory.cylinder s S ↔ (fun i => f ↑i) ∈ S
-- MeasureTheory.cylinder_empty
@[simp]
theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι)
Goal: MeasureTheory.cylinder s ∅ = ∅
-- MeasureTheory.cylinder_univ
@[simp]
theorem cylinder_univ (s : Finset ι) : cylinder s (univ : Set (∀ i : s, α i)) = univ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι)
Goal: MeasureTheory.cylinder s Set.univ = Set.univ
-- MeasureTheory.cylinder_eq_empty_iff
@[simp]
theorem cylinder_eq_empty_iff [h_nonempty : Nonempty (∀ i, α i)] (s : Finset ι)
    (S : Set (∀ i : s, α i)) :
    cylinder s S = ∅ ↔ S = ∅ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [h_nonempty : Nonempty ((i : ι) → α i)] (s : Finset ι) (S : Set ((i : { x // x ∈ s }) → α ↑i))
Goal: MeasureTheory.cylinder s S = ∅ ↔ S = ∅
-- MeasureTheory.inter_cylinder
theorem inter_cylinder (s₁ s₂ : Finset ι) (S₁ : Set (∀ i : s₁, α i)) (S₂ : Set (∀ i : s₂, α i))
    [DecidableEq ι] :
    cylinder s₁ S₁ ∩ cylinder s₂ S₂ =
      cylinder (s₁ ∪ s₂)
        ((fun f ↦ fun j : s₁ ↦ f ⟨j, Finset.mem_union_left s₂ j.prop⟩) ⁻¹' S₁ ∩
          (fun f ↦ fun j : s₂ ↦ f ⟨j, Finset.mem_union_right s₁ j.prop⟩) ⁻¹' S₂) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s₁ s₂ : Finset ι) (S₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)) (S₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)) [DecidableEq ι]
Goal: MeasureTheory.cylinder s₁ S₁ ∩ MeasureTheory.cylinder s₂ S₂ =
  MeasureTheory.cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∩ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)
-- MeasureTheory.inter_cylinder_same
theorem inter_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
    cylinder s S₁ ∩ cylinder s S₂ = cylinder s (S₁ ∩ S₂) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι) (S₁ S₂ : Set ((i : { x // x ∈ s }) → α ↑i))
Goal: MeasureTheory.cylinder s S₁ ∩ MeasureTheory.cylinder s S₂ = MeasureTheory.cylinder s (S₁ ∩ S₂)
-- MeasureTheory.union_cylinder
theorem union_cylinder (s₁ s₂ : Finset ι) (S₁ : Set (∀ i : s₁, α i)) (S₂ : Set (∀ i : s₂, α i))
    [DecidableEq ι] :
    cylinder s₁ S₁ ∪ cylinder s₂ S₂ =
      cylinder (s₁ ∪ s₂)
        ((fun f ↦ fun j : s₁ ↦ f ⟨j, Finset.mem_union_left s₂ j.prop⟩) ⁻¹' S₁ ∪
          (fun f ↦ fun j : s₂ ↦ f ⟨j, Finset.mem_union_right s₁ j.prop⟩) ⁻¹' S₂) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s₁ s₂ : Finset ι) (S₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)) (S₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)) [DecidableEq ι]
Goal: MeasureTheory.cylinder s₁ S₁ ∪ MeasureTheory.cylinder s₂ S₂ =
  MeasureTheory.cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)
-- MeasureTheory.union_cylinder_same
theorem union_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
    cylinder s S₁ ∪ cylinder s S₂ = cylinder s (S₁ ∪ S₂) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι) (S₁ S₂ : Set ((i : { x // x ∈ s }) → α ↑i))
Goal: MeasureTheory.cylinder s S₁ ∪ MeasureTheory.cylinder s S₂ = MeasureTheory.cylinder s (S₁ ∪ S₂)
-- MeasureTheory.compl_cylinder
theorem compl_cylinder (s : Finset ι) (S : Set (∀ i : s, α i)) :
    (cylinder s S)ᶜ = cylinder s (Sᶜ) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι) (S : Set ((i : { x // x ∈ s }) → α ↑i))
Goal: (MeasureTheory.cylinder s S)ᶜ = MeasureTheory.cylinder s Sᶜ
-- MeasureTheory.diff_cylinder_same
theorem diff_cylinder_same (s : Finset ι) (S T : Set (∀ i : s, α i)) :
    cylinder s S \ cylinder s T = cylinder s (S \ T) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} (s : Finset ι) (S T : Set ((i : { x // x ∈ s }) → α ↑i))
Goal: MeasureTheory.cylinder s S \ MeasureTheory.cylinder s T = MeasureTheory.cylinder s (S \ T)
-- MeasureTheory.eq_of_cylinder_eq_of_subset
theorem eq_of_cylinder_eq_of_subset [h_nonempty : Nonempty (∀ i, α i)] {I J : Finset ι}
    {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (h_eq : cylinder I S = cylinder J T)
    (hJI : J ⊆ I) :
    S = (fun f : ∀ i : I, α i ↦ fun j : J ↦ f ⟨j, hJI j.prop⟩) ⁻¹' T 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [h_nonempty : Nonempty ((i : ι) → α i)] {I J : Finset ι} {S : Set ((i : { x // x ∈ I }) → α ↑i)} {T : Set ((i : { x // x ∈ J }) → α ↑i)} (h_eq : MeasureTheory.cylinder I S = MeasureTheory.cylinder J T) (hJI : J ⊆ I)
Goal: S = (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' T
-- MeasureTheory.cylinder_eq_cylinder_union
theorem cylinder_eq_cylinder_union [DecidableEq ι] (I : Finset ι) (S : Set (∀ i : I, α i))
    (J : Finset ι) :
    cylinder I S =
      cylinder (I ∪ J) ((fun f ↦ fun j : I ↦ f ⟨j, Finset.mem_union_left J j.prop⟩) ⁻¹' S) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] (I : Finset ι) (S : Set ((i : { x // x ∈ I }) → α ↑i)) (J : Finset ι)
Goal: MeasureTheory.cylinder I S = MeasureTheory.cylinder (I ∪ J) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S)
-- MeasureTheory.disjoint_cylinder_iff
theorem disjoint_cylinder_iff [Nonempty (∀ i, α i)] {s t : Finset ι} {S : Set (∀ i : s, α i)}
    {T : Set (∀ i : t, α i)} [DecidableEq ι] :
    Disjoint (cylinder s S) (cylinder t T) ↔
      Disjoint
        ((fun f : ∀ i : (s ∪ t : Finset ι), α i
          ↦ fun j : s ↦ f ⟨j, Finset.mem_union_left t j.prop⟩) ⁻¹' S)
        ((fun f ↦ fun j : t ↦ f ⟨j, Finset.mem_union_right s j.prop⟩) ⁻¹' T) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [Nonempty ((i : ι) → α i)] {s t : Finset ι} {S : Set ((i : { x // x ∈ s }) → α ↑i)} {T : Set ((i : { x // x ∈ t }) → α ↑i)} [DecidableEq ι]
Goal: Disjoint (MeasureTheory.cylinder s S) (MeasureTheory.cylinder t T) ↔
  Disjoint ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' T)
-- MeasureTheory.IsClosed.cylinder
theorem IsClosed.cylinder [∀ i, TopologicalSpace (α i)] (s : Finset ι) {S : Set (∀ i : s, α i)}
    (hs : IsClosed S) : IsClosed (cylinder s S) 
Arguments: {ι : Type u_2} {α : ι → Type u_1} [(i : ι) → TopologicalSpace (α i)] (s : Finset ι) {S : Set ((i : { x // x ∈ s }) → α ↑i)} (hs : IsClosed S)
Goal: IsClosed (MeasureTheory.cylinder s S)
-- MeasurableSet.cylinder
theorem _root_.MeasurableSet.cylinder [∀ i, MeasurableSpace (α i)] (s : Finset ι)
    {S : Set (∀ i : s, α i)} (hS : MeasurableSet S) :
    MeasurableSet (cylinder s S) 
Arguments: {ι : Type u_2} {α : ι → Type u_1} [(i : ι) → MeasurableSpace (α i)] (s : Finset ι) {S : Set ((i : { x // x ∈ s }) → α ↑i)} (hS : MeasurableSet S)
Goal: MeasurableSet (MeasureTheory.cylinder s S)
-- MeasureTheory.empty_mem_measurableCylinders
theorem empty_mem_measurableCylinders (α : ι → Type*) [∀ i, MeasurableSpace (α i)] :
    ∅ ∈ measurableCylinders α 
Arguments: {ι : Type u_2} (α : ι → Type u_1) [(i : ι) → MeasurableSpace (α i)]
Goal: ∅ ∈ MeasureTheory.measurableCylinders α
-- MeasureTheory.mem_measurableCylinders
@[simp]
theorem mem_measurableCylinders (t : Set (∀ i, α i)) :
    t ∈ measurableCylinders α ↔ ∃ s S, MeasurableSet S ∧ t = cylinder s S 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] (t : Set ((i : ι) → α i))
Goal: t ∈ MeasureTheory.measurableCylinders α ↔ ∃ s S, MeasurableSet S ∧ t = MeasureTheory.cylinder s S
-- MeasureTheory.measurableCylinders.measurableSet
theorem measurableCylinders.measurableSet (ht : t ∈ measurableCylinders α) :
    MeasurableSet (measurableCylinders.set ht) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {t : Set ((i : ι) → α i)} (ht : t ∈ MeasureTheory.measurableCylinders α)
Goal: MeasurableSet (MeasureTheory.measurableCylinders.set ht)
-- MeasureTheory.measurableCylinders.eq_cylinder
theorem measurableCylinders.eq_cylinder (ht : t ∈ measurableCylinders α) :
    t = cylinder (measurableCylinders.finset ht) (measurableCylinders.set ht) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {t : Set ((i : ι) → α i)} (ht : t ∈ MeasureTheory.measurableCylinders α)
Goal: t = MeasureTheory.cylinder (MeasureTheory.measurableCylinders.finset ht) (MeasureTheory.measurableCylinders.set ht)