ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.CondIndepSets.iUnion
theorem CondIndepSets.iUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    (hyp : ∀ n, CondIndepSets m' hm' (s n) s' μ) :
    CondIndepSets m' hm' (⋃ n, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} (hyp : ∀ (n : ι), ProbabilityTheory.CondIndepSets m' hm' (s n) s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (⋃ n, s n) s' μ
-- ProbabilityTheory.CondIndepSets.bUnion
theorem CondIndepSets.bUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    {u : Set ι} (hyp : ∀ n ∈ u, CondIndepSets m' hm' (s n) s' μ) :
    CondIndepSets m' hm' (⋃ n ∈ u, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {u : Set ι} (hyp : ∀ n ∈ u, ProbabilityTheory.CondIndepSets m' hm' (s n) s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (⋃ n ∈ u, s n) s' μ
-- ProbabilityTheory.CondIndepSets.inter
theorem CondIndepSets.inter {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω))
    (h₁ : CondIndepSets m' hm' s₁ s' μ) :
    CondIndepSets m' hm' (s₁ ∩ s₂) s' μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) (h₁ : ProbabilityTheory.CondIndepSets m' hm' s₁ s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (s₁ ∩ s₂) s' μ
-- ProbabilityTheory.CondIndepSets.iInter
theorem CondIndepSets.iInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    (h : ∃ n, CondIndepSets m' hm' (s n) s' μ) :
    CondIndepSets m' hm' (⋂ n, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} (h : ∃ n, ProbabilityTheory.CondIndepSets m' hm' (s n) s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (⋂ n, s n) s' μ
-- ProbabilityTheory.CondIndepSets.bInter
theorem CondIndepSets.bInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    {u : Set ι} (h : ∃ n ∈ u, CondIndepSets m' hm' (s n) s' μ) :
    CondIndepSets m' hm' (⋂ n ∈ u, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {u : Set ι} (h : ∃ n ∈ u, ProbabilityTheory.CondIndepSets m' hm' (s n) s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (⋂ n ∈ u, s n) s' μ
-- ProbabilityTheory.condIndepSet_empty_right
theorem condIndepSet_empty_right (s : Set Ω) : CondIndepSet m' hm' s ∅ μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (s : Set Ω)
Goal: ProbabilityTheory.CondIndepSet m' hm' s ∅ μ
-- ProbabilityTheory.condIndepSet_empty_left
theorem condIndepSet_empty_left (s : Set Ω) : CondIndepSet m' hm' ∅ s μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (s : Set Ω)
Goal: ProbabilityTheory.CondIndepSet m' hm' ∅ s μ
-- ProbabilityTheory.CondIndep.symm
@[symm]
theorem CondIndep.symm {m' m₁ m₂ : MeasurableSpace Ω} {mΩ : MeasurableSpace Ω}
    [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : Measure Ω} [IsFiniteMeasure μ]
    (h : CondIndep m' m₁ m₂ hm' μ) :
    CondIndep m' m₂ m₁ hm' μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h : ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ)
Goal: ProbabilityTheory.CondIndep m' m₂ m₁ hm' μ
-- ProbabilityTheory.condIndep_bot_right
theorem condIndep_bot_right (m₁ : MeasurableSpace Ω) {m' : MeasurableSpace Ω}
    {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω]
    {hm' : m' ≤ mΩ} {μ : Measure Ω} [IsFiniteMeasure μ] :
    CondIndep m' m₁ ⊥ hm' μ 
Arguments: {Ω : Type u_1} (m₁ : MeasurableSpace Ω) {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndep m' m₁ ⊥ hm' μ
-- ProbabilityTheory.condIndep_bot_left
theorem condIndep_bot_left (m₁ : MeasurableSpace Ω) {m' : MeasurableSpace Ω}
    {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω]
    {hm' : m' ≤ mΩ} {μ : Measure Ω} [IsFiniteMeasure μ] :
    CondIndep m' ⊥ m₁ hm' μ 
Arguments: {Ω : Type u_1} (m₁ : MeasurableSpace Ω) {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndep m' ⊥ m₁ hm' μ
-- ProbabilityTheory.condIndep_of_condIndep_of_le_left
theorem condIndep_of_condIndep_of_le_left {m' m₁ m₂ m₃ : MeasurableSpace Ω}
    {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω]
    {hm' : m' ≤ mΩ} {μ : Measure Ω} [IsFiniteMeasure μ]
    (h_indep : CondIndep m' m₁ m₂ hm' μ) (h31 : m₃ ≤ m₁) :
    CondIndep m' m₃ m₂ hm' μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ m₃ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h_indep : ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ) (h31 : m₃ ≤ m₁)
Goal: ProbabilityTheory.CondIndep m' m₃ m₂ hm' μ
-- ProbabilityTheory.condIndep_of_condIndep_of_le_right
theorem condIndep_of_condIndep_of_le_right {m' m₁ m₂ m₃ : MeasurableSpace Ω}
    {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω]
    {hm' : m' ≤ mΩ} {μ : Measure Ω} [IsFiniteMeasure μ]
    (h_indep : CondIndep m' m₁ m₂ hm' μ) (h32 : m₃ ≤ m₂) :
    CondIndep m' m₁ m₃ hm' μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ m₃ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h_indep : ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ) (h32 : m₃ ≤ m₂)
Goal: ProbabilityTheory.CondIndep m' m₁ m₃ hm' μ
-- ProbabilityTheory.iCondIndepSets.condIndepSets
theorem iCondIndepSets.condIndepSets {s : ι → Set (Set Ω)}
    (h_indep : iCondIndepSets m' hm' s μ) {i j : ι} (hij : i ≠ j) :
    CondIndepSets m' hm' (s i) (s j) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} (h_indep : ProbabilityTheory.iCondIndepSets m' hm' s μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.CondIndepSets m' hm' (s i) (s j) μ
-- ProbabilityTheory.iCondIndep.condIndep
theorem iCondIndep.condIndep {m : ι → MeasurableSpace Ω}
    (h_indep : iCondIndep m' hm' m μ) {i j : ι} (hij : i ≠ j) :
      CondIndep m' (m i) (m j) hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : ι → MeasurableSpace Ω} (h_indep : ProbabilityTheory.iCondIndep m' hm' m μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.CondIndep m' (m i) (m j) hm' μ
-- ProbabilityTheory.iCondIndepFun.condIndepFun
theorem iCondIndepFun.condIndepFun {β : ι → Type*}
    {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i}
    (hf_Indep : iCondIndepFun m' hm' m f μ) {i j : ι} (hij : i ≠ j) :
    CondIndepFun m' hm' (f i) (f j) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_3} {m : (x : ι) → MeasurableSpace (β x)} {f : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' m f μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.CondIndepFun m' hm' (f i) (f j) μ
-- ProbabilityTheory.iCondIndep.iCondIndepSets
theorem iCondIndep.iCondIndepSets {m : ι → MeasurableSpace Ω}
    {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n))
    (h_indep : iCondIndep m' hm' m μ) :
    iCondIndepSets m' hm' s μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : ι → MeasurableSpace Ω} {s : ι → Set (Set Ω)} (hms : ∀ (n : ι), m n = MeasurableSpace.generateFrom (s n)) (h_indep : ProbabilityTheory.iCondIndep m' hm' m μ)
Goal: ProbabilityTheory.iCondIndepSets m' hm' s μ
-- ProbabilityTheory.CondIndep.condIndepSets
theorem CondIndep.condIndepSets {s1 s2 : Set (Set Ω)}
    (h_indep : CondIndep m' (generateFrom s1) (generateFrom s2) hm' μ) :
    CondIndepSets m' hm' s1 s2 μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s1 s2 : Set (Set Ω)} (h_indep : ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom s1) (MeasurableSpace.generateFrom s2) hm' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' s1 s2 μ
-- ProbabilityTheory.CondIndepSets.condIndep
theorem CondIndepSets.condIndep
    {p1 p2 : Set (Set Ω)} (h1 : m₁ ≤ mΩ) (h2 : m₂ ≤ mΩ)
    (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2)
    (hpm1 : m₁ = generateFrom p1) (hpm2 : m₂ = generateFrom p2)
    (hyp : CondIndepSets m' hm' p1 p2 μ) :
    CondIndep m' m₁ m₂ hm' μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {p1 p2 : Set (Set Ω)} (h1 : m₁ ≤ mΩ) (h2 : m₂ ≤ mΩ) (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hpm1 : m₁ = MeasurableSpace.generateFrom p1) (hpm2 : m₂ = MeasurableSpace.generateFrom p2) (hyp : ProbabilityTheory.CondIndepSets m' hm' p1 p2 μ)
Goal: ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ
-- ProbabilityTheory.CondIndepSets.condIndep'
theorem CondIndepSets.condIndep'
    {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
    (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : CondIndepSets m' hm' p1 p2 μ) :
    CondIndep m' (generateFrom p1) (generateFrom p2) hm' μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s) (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : ProbabilityTheory.CondIndepSets m' hm' p1 p2 μ)
Goal: ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom p1) (MeasurableSpace.generateFrom p2) hm' μ
-- ProbabilityTheory.condIndepSets_piiUnionInter_of_disjoint
theorem condIndepSets_piiUnionInter_of_disjoint {s : ι → Set (Set Ω)}
    {S T : Set ι} (h_indep : iCondIndepSets m' hm' s μ) (hST : Disjoint S T) :
    CondIndepSets m' hm' (piiUnionInter s S) (piiUnionInter s T) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set (Set Ω)} {S T : Set ι} (h_indep : ProbabilityTheory.iCondIndepSets m' hm' s μ) (hST : Disjoint S T)
Goal: ProbabilityTheory.CondIndepSets m' hm' (piiUnionInter s S) (piiUnionInter s T) μ
-- ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_of_disjoint
theorem iCondIndepSet.condIndep_generateFrom_of_disjoint {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s μ) (S T : Set ι)
    (hST : Disjoint S T) :
    CondIndep m' (generateFrom { t | ∃ n ∈ S, s n = t })
      (generateFrom { t | ∃ k ∈ T, s k = t }) hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iCondIndepSet m' hm' s μ) (S T : Set ι) (hST : Disjoint S T)
Goal: ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom {t | ∃ n ∈ S, s n = t})
  (MeasurableSpace.generateFrom {t | ∃ k ∈ T, s k = t}) hm' μ
-- ProbabilityTheory.condIndep_iSup_of_disjoint
theorem condIndep_iSup_of_disjoint {m : ι → MeasurableSpace Ω}
    (h_le : ∀ i, m i ≤ mΩ) (h_indep : iCondIndep m' hm' m μ) {S T : Set ι} (hST : Disjoint S T) :
    CondIndep m' (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : ι → MeasurableSpace Ω} (h_le : ∀ (i : ι), m i ≤ mΩ) (h_indep : ProbabilityTheory.iCondIndep m' hm' m μ) {S T : Set ι} (hST : Disjoint S T)
Goal: ProbabilityTheory.CondIndep m' (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) hm' μ
-- ProbabilityTheory.condIndep_iSup_of_directed_le
theorem condIndep_iSup_of_directed_le {m : ι → MeasurableSpace Ω}
    (h_indep : ∀ i, CondIndep m' (m i) m₁ hm' μ)
    (h_le : ∀ i, m i ≤ mΩ) (h_le' : m₁ ≤ mΩ) (hm : Directed (· ≤ ·) m) :
    CondIndep m' (⨆ i, m i) m₁ hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' m₁ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {m : ι → MeasurableSpace Ω} (h_indep : ∀ (i : ι), ProbabilityTheory.CondIndep m' (m i) m₁ hm' μ) (h_le : ∀ (i : ι), m i ≤ mΩ) (h_le' : m₁ ≤ mΩ) (hm : Directed (fun x x_1 => x ≤ x_1) m)
Goal: ProbabilityTheory.CondIndep m' (⨆ i, m i) m₁ hm' μ
-- ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_lt
theorem iCondIndepSet.condIndep_generateFrom_lt [Preorder ι] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s μ) (i : ι) :
    CondIndep m' (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [Preorder ι] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iCondIndepSet m' hm' s μ) (i : ι)
Goal: ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom {s i})
  (MeasurableSpace.generateFrom {t | ∃ j < i, s j = t}) hm' μ
-- ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_le
theorem iCondIndepSet.condIndep_generateFrom_le [LinearOrder ι] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s μ) (i : ι) {k : ι} (hk : i < k) :
    CondIndep m' (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [LinearOrder ι] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iCondIndepSet m' hm' s μ) (i : ι) {k : ι} (hk : i < k)
Goal: ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom {s k})
  (MeasurableSpace.generateFrom {t | ∃ j ≤ i, s j = t}) hm' μ
-- ProbabilityTheory.iCondIndepSet.condIndep_generateFrom_le_nat
theorem iCondIndepSet.condIndep_generateFrom_le_nat {s : ℕ → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iCondIndepSet m' hm' s μ) (n : ℕ) :
    CondIndep m' (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) hm' μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s : ℕ → Set Ω} (hsm : ∀ (n : ℕ), MeasurableSet (s n)) (hs : ProbabilityTheory.iCondIndepSet m' hm' s μ) (n : ℕ)
Goal: ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom {s (n + 1)})
  (MeasurableSpace.generateFrom {t | ∃ k ≤ n, s k = t}) hm' μ
-- ProbabilityTheory.condIndep_iSup_of_monotone
theorem condIndep_iSup_of_monotone [SemilatticeSup ι] {m : ι → MeasurableSpace Ω}
    (h_indep : ∀ i, CondIndep m' (m i) m₁ hm' μ) (h_le : ∀ i, m i ≤ mΩ) (h_le' : m₁ ≤ mΩ)
    (hm : Monotone m) :
    CondIndep m' (⨆ i, m i) m₁ hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' m₁ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [SemilatticeSup ι] {m : ι → MeasurableSpace Ω} (h_indep : ∀ (i : ι), ProbabilityTheory.CondIndep m' (m i) m₁ hm' μ) (h_le : ∀ (i : ι), m i ≤ mΩ) (h_le' : m₁ ≤ mΩ) (hm : Monotone m)
Goal: ProbabilityTheory.CondIndep m' (⨆ i, m i) m₁ hm' μ
-- ProbabilityTheory.condIndep_iSup_of_antitone
theorem condIndep_iSup_of_antitone [SemilatticeInf ι] {m : ι → MeasurableSpace Ω}
    (h_indep : ∀ i, CondIndep m' (m i) m₁ hm' μ) (h_le : ∀ i, m i ≤ mΩ) (h_le' : m₁ ≤ mΩ)
    (hm : Antitone m) :
    CondIndep m' (⨆ i, m i) m₁ hm' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' m₁ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] [SemilatticeInf ι] {m : ι → MeasurableSpace Ω} (h_indep : ∀ (i : ι), ProbabilityTheory.CondIndep m' (m i) m₁ hm' μ) (h_le : ∀ (i : ι), m i ≤ mΩ) (h_le' : m₁ ≤ mΩ) (hm : Antitone m)
Goal: ProbabilityTheory.CondIndep m' (⨆ i, m i) m₁ hm' μ
-- ProbabilityTheory.iCondIndepSets.piiUnionInter_of_not_mem
theorem iCondIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
    (hp_ind : iCondIndepSets m' hm' π μ) (haS : a ∉ S) :
    CondIndepSets m' hm' (piiUnionInter π S) (π a) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι} (hp_ind : ProbabilityTheory.iCondIndepSets m' hm' π μ) (haS : a ∉ S)
Goal: ProbabilityTheory.CondIndepSets m' hm' (piiUnionInter π ↑S) (π a) μ
-- ProbabilityTheory.iCondIndepSets.iCondIndep
theorem iCondIndepSets.iCondIndep (m : ι → MeasurableSpace Ω)
    (h_le : ∀ i, m i ≤ mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
    (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iCondIndepSets m' hm' π μ) :
    iCondIndep m' hm' m μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (m : ι → MeasurableSpace Ω) (h_le : ∀ (i : ι), m i ≤ mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ (n : ι), IsPiSystem (π n)) (h_generate : ∀ (i : ι), m i = MeasurableSpace.generateFrom (π i)) (h_ind : ProbabilityTheory.iCondIndepSets m' hm' π μ)
Goal: ProbabilityTheory.iCondIndep m' hm' m μ
-- ProbabilityTheory.CondIndepSets.condIndepSet_of_mem
theorem CondIndepSets.condIndepSet_of_mem (hs : s ∈ S) (ht : t ∈ T)
    (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (μ : Measure Ω) [IsFiniteMeasure μ]
    (h_indep : CondIndepSets m' hm' S T μ) :
    CondIndepSet m' hm' s t μ 
Arguments: {Ω : Type u_1} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {s t : Set Ω} (S T : Set (Set Ω)) (hs : s ∈ S) (ht : t ∈ T) (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (h_indep : ProbabilityTheory.CondIndepSets m' hm' S T μ)
Goal: ProbabilityTheory.CondIndepSet m' hm' s t μ
-- ProbabilityTheory.CondIndep.condIndepSet_of_measurableSet
theorem CondIndep.condIndepSet_of_measurableSet {μ : Measure Ω} [IsFiniteMeasure μ]
    (h_indep : CondIndep m' m₁ m₂ hm' μ) {s t : Set Ω} (hs : MeasurableSet[m₁] s)
    (ht : MeasurableSet[m₂] t) :
    CondIndepSet m' hm' s t μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (h_indep : ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ) {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ProbabilityTheory.CondIndepSet m' hm' s t μ
-- ProbabilityTheory.condIndep_iff_forall_condIndepSet
theorem condIndep_iff_forall_condIndepSet (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndep m' m₁ m₂ hm' μ ↔ ∀ s t, MeasurableSet[m₁] s → MeasurableSet[m₂] t
      → CondIndepSet m' hm' s t μ 
Arguments: {Ω : Type u_1} {m' m₁ m₂ mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ ↔
  ∀ (s t : Set Ω), MeasurableSet s → MeasurableSet t → ProbabilityTheory.CondIndepSet m' hm' s t μ
-- ProbabilityTheory.condIndepFun_iff_condexp_inter_preimage_eq_mul
theorem condIndepFun_iff_condexp_inter_preimage_eq_mul {mβ : MeasurableSpace β}
    {mβ' : MeasurableSpace β'} (hf : Measurable f) (hg : Measurable g) :
    CondIndepFun m' hm' f g μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t
        → (μ⟦f ⁻¹' s ∩ g ⁻¹' t | m'⟧) =ᵐ[μ] fun ω ↦ (μ⟦f ⁻¹' s | m'⟧) ω * (μ⟦g ⁻¹' t | m'⟧) ω 
Arguments: {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} (hf : Measurable f) (hg : Measurable g)
Goal: ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s →
      MeasurableSet t →
        μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω => 1|m'] =ᵐ[μ] fun ω =>
          (μ[(f ⁻¹' s).indicator fun ω => 1|m']) ω * (μ[(g ⁻¹' t).indicator fun ω => 1|m']) ω
-- ProbabilityTheory.iCondIndepFun_iff_condexp_inter_preimage_eq_mul
theorem iCondIndepFun_iff_condexp_inter_preimage_eq_mul {β : ι → Type*}
    (m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) (hf : ∀ i, Measurable (f i)) :
    iCondIndepFun m' hm' m f μ ↔
      ∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
        (μ⟦⋂ i ∈ S, f i ⁻¹' sets i| m'⟧) =ᵐ[μ] ∏ i ∈ S, (μ⟦f i ⁻¹' sets i | m'⟧) 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {β : ι → Type u_5} (m : (x : ι) → MeasurableSpace (β x)) (f : (i : ι) → Ω → β i) (hf : ∀ (i : ι), Measurable (f i))
Goal: ProbabilityTheory.iCondIndepFun m' hm' m f μ ↔
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) →
      μ[(⋂ i ∈ S, f i ⁻¹' sets i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ S, μ[(f i ⁻¹' sets i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndepFun_iff_condIndepSet_preimage
theorem condIndepFun_iff_condIndepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    (hf : Measurable f) (hg : Measurable g) :
    CondIndepFun m' hm' f g μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t → CondIndepSet m' hm' (f ⁻¹' s) (g ⁻¹' t) μ 
Arguments: {Ω : Type u_1} {β : Type u_3} {β' : Type u_4} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} (hf : Measurable f) (hg : Measurable g)
Goal: ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s → MeasurableSet t → ProbabilityTheory.CondIndepSet m' hm' (f ⁻¹' s) (g ⁻¹' t) μ
-- ProbabilityTheory.CondIndepFun.symm
@[symm]
nonrec theorem CondIndepFun.symm {_ : MeasurableSpace β} {f g : Ω → β}
    (hfg : CondIndepFun m' hm' f g μ) :
    CondIndepFun m' hm' g f μ 
Arguments: {Ω : Type u_1} {β : Type u_3} {m' mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] {hm' : m' ≤ mΩ} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ]
Goal: ∀ {x : MeasurableSpace β} {f g : Ω → β},
  ProbabilityTheory.CondIndepFun m' hm' f g μ → ProbabilityTheory.CondIndepFun m' hm' g f μ