ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.hasPDF_iff_of_aemeasurable
nonrec theorem _root_.Real.hasPDF_iff_of_aemeasurable (hX : AEMeasurable X ℙ) :
    HasPDF X ℙ ↔ map X ℙ ≪ volume 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ} (hX : AEMeasurable X ℙ)
Goal: MeasureTheory.HasPDF X ℙ MeasureTheory.volume ↔ MeasureTheory.Measure.map X ℙ ≪ MeasureTheory.volume
-- Real.hasPDF_iff
theorem _root_.Real.hasPDF_iff : HasPDF X ℙ ↔ AEMeasurable X ℙ ∧ map X ℙ ≪ volume 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ}
Goal: MeasureTheory.HasPDF X ℙ MeasureTheory.volume ↔ AEMeasurable X ℙ ∧ MeasureTheory.Measure.map X ℙ ≪ MeasureTheory.volume
-- MeasureTheory.pdf.integral_mul_eq_integral
theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x).toReal = ∫ x, X x ∂ℙ 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ} [MeasureTheory.HasPDF X ℙ MeasureTheory.volume]
Goal: ∫ (x : ℝ), x * (MeasureTheory.pdf X ℙ MeasureTheory.volume x).toReal = ∫ (x : Ω), X x ∂ℙ
-- MeasureTheory.pdf.hasFiniteIntegral_mul
theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
    (hgi : ∫⁻ x, ‖f x‖₊ * g x ≠ ∞) :
    HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal 
Arguments: {Ω : Type u_1} {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure ℙ] {X : Ω → ℝ} {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : MeasureTheory.pdf X ℙ MeasureTheory.volume =ᵐ[MeasureTheory.volume] g) (hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤)
Goal: MeasureTheory.HasFiniteIntegral (fun x => f x * (MeasureTheory.pdf X ℙ MeasureTheory.volume x).toReal)
  MeasureTheory.volume
-- MeasureTheory.pdf.indepFun_iff_pdf_prod_eq_pdf_mul_pdf
theorem indepFun_iff_pdf_prod_eq_pdf_mul_pdf
    [IsFiniteMeasure ℙ] [SigmaFinite μ] [SigmaFinite ν] [HasPDF (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν)] :
    IndepFun X Y ℙ ↔
      pdf (fun ω ↦ (X ω, Y ω)) ℙ (μ.prod ν) =ᵐ[μ.prod ν] fun z ↦ pdf X ℙ μ z.1 * pdf Y ℙ ν z.2 
Arguments: {Ω : Type u_1} {E : Type u_2} [MeasurableSpace E] {m : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {μ : MeasureTheory.Measure E} {F : Type u_3} [MeasurableSpace F] {ν : MeasureTheory.Measure F} {X : Ω → E} {Y : Ω → F} [MeasureTheory.IsFiniteMeasure ℙ] [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [MeasureTheory.HasPDF (fun ω => (X ω, Y ω)) ℙ (μ.prod ν)]
Goal: ProbabilityTheory.IndepFun X Y ℙ ↔
  MeasureTheory.pdf (fun ω => (X ω, Y ω)) ℙ (μ.prod ν) =ᵐ[μ.prod ν] fun z =>
    MeasureTheory.pdf X ℙ μ z.1 * MeasureTheory.pdf Y ℙ ν z.2
-- ProbabilityTheory.Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self
theorem Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
    (h_indep : Kernel.IndepSet t t κ μα) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 ∨ κ a t = ∞ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} {t : Set Ω} (h_indep : ProbabilityTheory.Kernel.IndepSet t t κ μα)
Goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤
-- ProbabilityTheory.measure_eq_zero_or_one_or_top_of_indepSet_self
theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 ∨ μ t = ∞ 
Arguments: {Ω : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {t : Set Ω} (h_indep : ProbabilityTheory.IndepSet t t μ)
Goal: μ t = 0 ∨ μ t = 1 ∨ μ t = ⊤
-- ProbabilityTheory.Kernel.measure_eq_zero_or_one_of_indepSet_self
theorem Kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
    (h_indep : IndepSet t t κ μα) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [∀ (a : α), MeasureTheory.IsFiniteMeasure (κ a)] {t : Set Ω} (h_indep : ProbabilityTheory.Kernel.IndepSet t t κ μα)
Goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1
-- ProbabilityTheory.measure_eq_zero_or_one_of_indepSet_self
theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
    (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 
Arguments: {Ω : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {t : Set Ω} (h_indep : ProbabilityTheory.IndepSet t t μ)
Goal: μ t = 0 ∨ μ t = 1
-- ProbabilityTheory.condexp_eq_zero_or_one_of_condIndepSet_self
theorem condexp_eq_zero_or_one_of_condIndepSet_self
    [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [hμ : IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t)
    (h_indep : CondIndepSet m hm t t μ) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
Arguments: {Ω : Type u_2} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [hμ : MeasureTheory.IsFiniteMeasure μ] {t : Set Ω} (ht : MeasurableSet t) (h_indep : ProbabilityTheory.CondIndepSet m hm t t μ)
Goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1
-- ProbabilityTheory.Kernel.indep_biSup_compl
theorem Kernel.indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) (t : Set ι) :
    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα) (t : Set ι)
Goal: ProbabilityTheory.Kernel.Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα
-- ProbabilityTheory.indep_biSup_compl
theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
    Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (t : Set ι)
Goal: ProbabilityTheory.Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ
-- ProbabilityTheory.condIndep_biSup_compl
theorem condIndep_biSup_compl [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (t : Set ι) :
    CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (t : Set ι)
Goal: ProbabilityTheory.CondIndep m (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) hm μ
-- ProbabilityTheory.Kernel.indep_biSup_limsup
theorem Kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) {t : Set ι} (ht : p t) :
    Indep (⨆ n ∈ t, s n) (limsup s f) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {p : Set ι → Prop} {f : Filter ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) {t : Set ι} (ht : p t)
Goal: ProbabilityTheory.Kernel.Indep (⨆ n ∈ t, s n) (Filter.limsup s f) κ μα
-- ProbabilityTheory.indep_biSup_limsup
theorem indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    {t : Set ι} (ht : p t) :
    Indep (⨆ n ∈ t, s n) (limsup s f) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {p : Set ι → Prop} {f : Filter ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) {t : Set ι} (ht : p t)
Goal: ProbabilityTheory.Indep (⨆ n ∈ t, s n) (Filter.limsup s f) μ
-- ProbabilityTheory.condIndep_biSup_limsup
theorem condIndep_biSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    {t : Set ι} (ht : p t) :
    CondIndep m (⨆ n ∈ t, s n) (limsup s f) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {p : Set ι → Prop} {f : Filter ι} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) {t : Set ι} (ht : p t)
Goal: ProbabilityTheory.CondIndep m (⨆ n ∈ t, s n) (Filter.limsup s f) hm μ
-- ProbabilityTheory.Kernel.indep_iSup_directed_limsup
theorem Kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) → ProbabilityTheory.Kernel.Indep (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) κ μα
-- ProbabilityTheory.indep_iSup_directed_limsup
theorem indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a))
Goal: ProbabilityTheory.Indep (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) μ
-- ProbabilityTheory.condIndep_iSup_directed_limsup
theorem condIndep_iSup_directed_limsup [StandardBorelSpace Ω]
    [Nonempty Ω] (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
    CondIndep m (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a))
Goal: ProbabilityTheory.CondIndep m (⨆ a, ⨆ n ∈ ns a, s n) (Filter.limsup s f) hm μ
-- ProbabilityTheory.Kernel.indep_iSup_limsup
theorem Kernel.indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (⨆ n, s n) (limsup s f) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ProbabilityTheory.Kernel.Indep (⨆ n, s n) (Filter.limsup s f) κ μα
-- ProbabilityTheory.indep_iSup_limsup
theorem indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (⨆ n, s n) (limsup s f) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a)
Goal: ProbabilityTheory.Indep (⨆ n, s n) (Filter.limsup s f) μ
-- ProbabilityTheory.condIndep_iSup_limsup
theorem condIndep_iSup_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    CondIndep m (⨆ n, s n) (limsup s f) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a)
Goal: ProbabilityTheory.CondIndep m (⨆ n, s n) (Filter.limsup s f) hm μ
-- ProbabilityTheory.Kernel.indep_limsup_self
theorem Kernel.indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (limsup s f) (limsup s f) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ProbabilityTheory.Kernel.Indep (Filter.limsup s f) (Filter.limsup s f) κ μα
-- ProbabilityTheory.indep_limsup_self
theorem indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    Indep (limsup s f) (limsup s f) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a)
Goal: ProbabilityTheory.Indep (Filter.limsup s f) (Filter.limsup s f) μ
-- ProbabilityTheory.condIndep_limsup_self
theorem condIndep_limsup_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) (hf : ∀ t, p t → tᶜ ∈ f)
    (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
    CondIndep m (limsup s f) (limsup s f) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a)
Goal: ProbabilityTheory.CondIndep m (Filter.limsup s f) (Filter.limsup s f) hm μ
-- ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ]
Goal: ∀ {α_1 : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α_1 → Set ι},
  (∀ (n : ι), s n ≤ m0) →
    ProbabilityTheory.Kernel.iIndep s κ μα →
      (∀ (t : Set ι), p t → tᶜ ∈ f) →
        Directed (fun x x_1 => x ≤ x_1) ns →
          (∀ (a : α_1), p (ns a)) →
            (∀ (n : ι), ∃ a, n ∈ ns a) → ∀ {t : Set Ω}, MeasurableSet t → ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1
-- ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup
theorem measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    μ t = 0 ∨ μ t = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: μ t = 0 ∨ μ t = 1
-- ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup
theorem condexp_zero_or_one_of_measurableSet_limsup [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ)
    (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
    (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {α : Type u_4} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι} [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) (hf : ∀ (t : Set ι), p t → tᶜ ∈ f) (hns : Directed (fun x x_1 => x ≤ x_1) ns) (hnsp : ∀ (a : α), p (ns a)) (hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1
-- ProbabilityTheory.Kernel.indep_limsup_atTop_self
theorem Kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) :
    Indep (limsup s atTop) (limsup s atTop) κ μα 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα)
Goal: ProbabilityTheory.Kernel.Indep (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) κ μα
-- ProbabilityTheory.indep_limsup_atTop_self
theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
    Indep (limsup s atTop) (limsup s atTop) μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ)
Goal: ProbabilityTheory.Indep (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) μ
-- ProbabilityTheory.condIndep_limsup_atTop_self
theorem condIndep_limsup_atTop_self [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ]
    (h_le : ∀ n, s n ≤ m0) (h_indep : iCondIndep m hm s μ) :
    CondIndep m (limsup s atTop) (limsup s atTop) hm μ 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ)
Goal: ProbabilityTheory.CondIndep m (Filter.limsup s Filter.atTop) (Filter.limsup s Filter.atTop) hm μ
-- ProbabilityTheory.Kernel.measure_zero_or_one_of_measurableSet_limsup_atTop
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μα : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.Kernel.iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: ∀ᵐ (a : α) ∂μα, (κ a) t = 0 ∨ (κ a) t = 1
-- ProbabilityTheory.measure_zero_or_one_of_measurableSet_limsup_atTop
theorem measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
    (h_indep : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    μ t = 0 ∨ μ t = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: μ t = 0 ∨ μ t = 1
-- ProbabilityTheory.condexp_zero_or_one_of_measurableSet_limsup_atTop
theorem condexp_zero_or_one_of_measurableSet_limsup_atTop [StandardBorelSpace Ω] [Nonempty Ω]
    (hm : m ≤ m0) [IsFiniteMeasure μ] (h_le : ∀ n, s n ≤ m0)
    (h_indep : iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
    ∀ᵐ ω ∂μ, (μ⟦t | m⟧) ω = 0 ∨ (μ⟦t | m⟧) ω = 1 
Arguments: {Ω : Type u_2} {ι : Type u_3} {s : ι → MeasurableSpace Ω} {m m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeSup ι] [NoMaxOrder ι] [Nonempty ι] [StandardBorelSpace Ω] [Nonempty Ω] (hm : m ≤ m0) [MeasureTheory.IsFiniteMeasure μ] (h_le : ∀ (n : ι), s n ≤ m0) (h_indep : ProbabilityTheory.iCondIndep m hm s μ) {t : Set Ω} (ht_tail : MeasurableSet t)
Goal: ∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1