ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.iIndep.iIndepSets
theorem iIndep.iIndepSets {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} {m : ι → MeasurableSpace Ω}
    {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : iIndep m κ μ) :
    iIndepSets s κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {m : ι → MeasurableSpace Ω} {s : ι → Set (Set Ω)} (hms : ∀ (n : ι), m n = MeasurableSpace.generateFrom (s n)) (h_indep : ProbabilityTheory.Kernel.iIndep m κ μ)
Goal: ProbabilityTheory.Kernel.iIndepSets s κ μ
-- ProbabilityTheory.Kernel.Indep.indepSets
theorem Indep.indepSets {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} {s1 s2 : Set (Set Ω)}
    (h_indep : Indep (generateFrom s1) (generateFrom s2) κ μ) :
    IndepSets s1 s2 κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {s1 s2 : Set (Set Ω)} (h_indep : ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom s1) (MeasurableSpace.generateFrom s2) κ μ)
Goal: ProbabilityTheory.Kernel.IndepSets s1 s2 κ μ
-- ProbabilityTheory.Kernel.IndepSets.indep_aux
theorem IndepSets.indep_aux {m₂ m : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h2 : m₂ ≤ m)
    (hp2 : IsPiSystem p2) (hpm2 : m₂ = generateFrom p2) (hyp : IndepSets p1 p2 κ μ) {t1 t2 : Set Ω}
    (ht1 : t1 ∈ p1) (ht1m : MeasurableSet[m] t1) (ht2m : MeasurableSet[m₂] t2) :
    ∀ᵐ a ∂μ, κ a (t1 ∩ t2) = κ a t1 * κ a t2 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₂ m : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h2 : m₂ ≤ m) (hp2 : IsPiSystem p2) (hpm2 : m₂ = MeasurableSpace.generateFrom p2) (hyp : ProbabilityTheory.Kernel.IndepSets p1 p2 κ μ) {t1 t2 : Set Ω} (ht1 : t1 ∈ p1) (ht1m : MeasurableSet t1) (ht2m : MeasurableSet t2)
Goal: ∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2
-- ProbabilityTheory.Kernel.IndepSets.indep
theorem IndepSets.indep {m1 m2 m : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α}
    [IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
    (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
    (hyp : IndepSets p1 p2 κ μ) :
    Indep m1 m2 κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m1 m2 m : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hpm1 : m1 = MeasurableSpace.generateFrom p1) (hpm2 : m2 = MeasurableSpace.generateFrom p2) (hyp : ProbabilityTheory.Kernel.IndepSets p1 p2 κ μ)
Goal: ProbabilityTheory.Kernel.Indep m1 m2 κ μ
-- ProbabilityTheory.Kernel.IndepSets.indep'
theorem IndepSets.indep' {_mΩ : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
    {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
    (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 κ μ) :
    Indep (generateFrom p1) (generateFrom p2) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s) (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : ProbabilityTheory.Kernel.IndepSets p1 p2 κ μ)
Goal: ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom p1) (MeasurableSpace.generateFrom p2) κ μ
-- ProbabilityTheory.Kernel.indepSets_piiUnionInter_of_disjoint
theorem indepSets_piiUnionInter_of_disjoint [IsMarkovKernel κ] {s : ι → Set (Set Ω)}
    {S T : Set ι} (h_indep : iIndepSets s κ μ) (hST : Disjoint S T) :
    IndepSets (piiUnionInter s S) (piiUnionInter s T) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {s : ι → Set (Set Ω)} {S T : Set ι} (h_indep : ProbabilityTheory.Kernel.iIndepSets s κ μ) (hST : Disjoint S T)
Goal: ProbabilityTheory.Kernel.IndepSets (piiUnionInter s S) (piiUnionInter s T) κ μ
-- ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_of_disjoint
theorem iIndepSet.indep_generateFrom_of_disjoint [IsMarkovKernel κ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (S T : Set ι) (hST : Disjoint S T) :
    Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.Kernel.iIndepSet s κ μ) (S T : Set ι) (hST : Disjoint S T)
Goal: ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom {t | ∃ n ∈ S, s n = t})
  (MeasurableSpace.generateFrom {t | ∃ k ∈ T, s k = t}) κ μ
-- ProbabilityTheory.Kernel.indep_iSup_of_disjoint
theorem indep_iSup_of_disjoint [IsMarkovKernel κ] {m : ι → MeasurableSpace Ω}
    (h_le : ∀ i, m i ≤ _mΩ) (h_indep : iIndep m κ μ) {S T : Set ι} (hST : Disjoint S T) :
    Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {m : ι → MeasurableSpace Ω} (h_le : ∀ (i : ι), m i ≤ _mΩ) (h_indep : ProbabilityTheory.Kernel.iIndep m κ μ) {S T : Set ι} (hST : Disjoint S T)
Goal: ProbabilityTheory.Kernel.Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) κ μ
-- ProbabilityTheory.Kernel.indep_iSup_of_directed_le
theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] (h_indep : ∀ i, Indep (m i) m' κ μ)
    (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) :
    Indep (⨆ i, m i) m' κ μ 
Arguments: {α : Type u_1} {ι : Type u_3} {_mα : MeasurableSpace α} {Ω : Type u_4} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (h_indep : ∀ (i : ι), ProbabilityTheory.Kernel.Indep (m i) m' κ μ) (h_le : ∀ (i : ι), m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (fun x x_1 => x ≤ x_1) m)
Goal: ProbabilityTheory.Kernel.Indep (⨆ i, m i) m' κ μ
-- ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_lt
theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsMarkovKernel κ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (i : ι) :
    Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [Preorder ι] [ProbabilityTheory.IsMarkovKernel κ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.Kernel.iIndepSet s κ μ) (i : ι)
Goal: ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom {s i})
  (MeasurableSpace.generateFrom {t | ∃ j < i, s j = t}) κ μ
-- ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_le
theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsMarkovKernel κ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (i : ι) {k : ι} (hk : i < k) :
    Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [LinearOrder ι] [ProbabilityTheory.IsMarkovKernel κ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.Kernel.iIndepSet s κ μ) (i : ι) {k : ι} (hk : i < k)
Goal: ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom {s k})
  (MeasurableSpace.generateFrom {t | ∃ j ≤ i, s j = t}) κ μ
-- ProbabilityTheory.Kernel.iIndepSet.indep_generateFrom_le_nat
theorem iIndepSet.indep_generateFrom_le_nat [IsMarkovKernel κ] {s : ℕ → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (n : ℕ) :
    Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] {s : ℕ → Set Ω} (hsm : ∀ (n : ℕ), MeasurableSet (s n)) (hs : ProbabilityTheory.Kernel.iIndepSet s κ μ) (n : ℕ)
Goal: ProbabilityTheory.Kernel.Indep (MeasurableSpace.generateFrom {s (n + 1)})
  (MeasurableSpace.generateFrom {t | ∃ k ≤ n, s k = t}) κ μ
-- ProbabilityTheory.Kernel.indep_iSup_of_monotone
theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
    (h_indep : ∀ i, Indep (m i) m' κ μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
    (hm : Monotone m) :
    Indep (⨆ i, m i) m' κ μ 
Arguments: {α : Type u_1} {ι : Type u_3} {_mα : MeasurableSpace α} [SemilatticeSup ι] {Ω : Type u_4} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (h_indep : ∀ (i : ι), ProbabilityTheory.Kernel.Indep (m i) m' κ μ) (h_le : ∀ (i : ι), m i ≤ m0) (h_le' : m' ≤ m0) (hm : Monotone m)
Goal: ProbabilityTheory.Kernel.Indep (⨆ i, m i) m' κ μ
-- ProbabilityTheory.Kernel.indep_iSup_of_antitone
theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
    (h_indep : ∀ i, Indep (m i) m' κ μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
    (hm : Antitone m) :
    Indep (⨆ i, m i) m' κ μ 
Arguments: {α : Type u_1} {ι : Type u_3} {_mα : MeasurableSpace α} [SemilatticeInf ι] {Ω : Type u_4} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (h_indep : ∀ (i : ι), ProbabilityTheory.Kernel.Indep (m i) m' κ μ) (h_le : ∀ (i : ι), m i ≤ m0) (h_le' : m' ≤ m0) (hm : Antitone m)
Goal: ProbabilityTheory.Kernel.Indep (⨆ i, m i) m' κ μ
-- ProbabilityTheory.Kernel.iIndepSets.piiUnionInter_of_not_mem
theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι}
    (hp_ind : iIndepSets π κ μ) (haS : a ∉ S) :
    IndepSets (piiUnionInter π S) (π a) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι} (hp_ind : ProbabilityTheory.Kernel.iIndepSets π κ μ) (haS : a ∉ S)
Goal: ProbabilityTheory.Kernel.IndepSets (piiUnionInter π ↑S) (π a) κ μ
-- ProbabilityTheory.Kernel.iIndepSets.iIndep
theorem iIndepSets.iIndep [IsMarkovKernel κ] (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 : iIndepSets π κ μ) :
    iIndep m κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} [ProbabilityTheory.IsMarkovKernel κ] (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.Kernel.iIndepSets π κ μ)
Goal: ProbabilityTheory.Kernel.iIndep m κ μ
-- ProbabilityTheory.Kernel.iIndepSet_iff_iIndepSets_singleton
theorem iIndepSet_iff_iIndepSets_singleton {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
    [IsMarkovKernel κ] {μ : Measure α} {f : ι → Set Ω}
    (hf : ∀ i, MeasurableSet (f i)) :
    iIndepSet f κ μ ↔ iIndepSets (fun i ↦ {f i}) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure α} {f : ι → Set Ω} (hf : ∀ (i : ι), MeasurableSet (f i))
Goal: ProbabilityTheory.Kernel.iIndepSet f κ μ ↔ ProbabilityTheory.Kernel.iIndepSets (fun i => {f i}) κ μ
-- ProbabilityTheory.Kernel.iIndepSet_iff_meas_biInter
theorem iIndepSet_iff_meas_biInter {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
    [IsMarkovKernel κ] {μ : Measure α} {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
    iIndepSet f κ μ ↔ ∀ s, ∀ᵐ a ∂μ, κ a (⋂ i ∈ s, f i) = ∏ i ∈ s, κ a (f i) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure α} {f : ι → Set Ω} (hf : ∀ (i : ι), MeasurableSet (f i))
Goal: ProbabilityTheory.Kernel.iIndepSet f κ μ ↔ ∀ (s : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)
-- ProbabilityTheory.Kernel.iIndepSets.iIndepSet_of_mem
theorem iIndepSets.iIndepSet_of_mem {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
    [IsMarkovKernel κ] {μ : Measure α} {π : ι → Set (Set Ω)} {f : ι → Set Ω}
    (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
    (hπ : iIndepSets π κ μ) :
    iIndepSet f κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure α} {π : ι → Set (Set Ω)} {f : ι → Set Ω} (hfπ : ∀ (i : ι), f i ∈ π i) (hf : ∀ (i : ι), MeasurableSet (f i)) (hπ : ProbabilityTheory.Kernel.iIndepSets π κ μ)
Goal: ProbabilityTheory.Kernel.iIndepSet f κ μ
-- ProbabilityTheory.Kernel.indepSet_iff_indepSets_singleton
theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
    (ht_meas : MeasurableSet t) (κ : Kernel α Ω) (μ : Measure α)
    [IsMarkovKernel κ] :
    IndepSet s t κ μ ↔ IndepSets {s} {t} κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s t : Set Ω} {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (κ : ProbabilityTheory.Kernel α Ω) (μ : MeasureTheory.Measure α) [ProbabilityTheory.IsMarkovKernel κ]
Goal: ProbabilityTheory.Kernel.IndepSet s t κ μ ↔ ProbabilityTheory.Kernel.IndepSets {s} {t} κ μ
-- ProbabilityTheory.Kernel.indepSet_iff_measure_inter_eq_mul
theorem indepSet_iff_measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
    (ht_meas : MeasurableSet t) (κ : Kernel α Ω) (μ : Measure α)
    [IsMarkovKernel κ] :
    IndepSet s t κ μ ↔ ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s t : Set Ω} {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (κ : ProbabilityTheory.Kernel α Ω) (μ : MeasureTheory.Measure α) [ProbabilityTheory.IsMarkovKernel κ]
Goal: ProbabilityTheory.Kernel.IndepSet s t κ μ ↔ ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t
-- ProbabilityTheory.Kernel.IndepSets.indepSet_of_mem
theorem IndepSets.indepSet_of_mem {_m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
    (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
    (κ : Kernel α Ω) (μ : Measure α) [IsMarkovKernel κ]
    (h_indep : IndepSets S T κ μ) :
    IndepSet s t κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {s t : Set Ω} (S T : Set (Set Ω)) {_m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T) (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (κ : ProbabilityTheory.Kernel α Ω) (μ : MeasureTheory.Measure α) [ProbabilityTheory.IsMarkovKernel κ] (h_indep : ProbabilityTheory.Kernel.IndepSets S T κ μ)
Goal: ProbabilityTheory.Kernel.IndepSet s t κ μ
-- ProbabilityTheory.Kernel.Indep.indepSet_of_measurableSet
theorem Indep.indepSet_of_measurableSet {m₁ m₂ m0 : MeasurableSpace Ω} {κ : Kernel α Ω}
    {μ : Measure α}
    (h_indep : Indep m₁ m₂ κ μ) {s t : Set Ω} (hs : MeasurableSet[m₁] s)
    (ht : MeasurableSet[m₂] t) :
    IndepSet s t κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {m₁ m₂ m0 : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} (h_indep : ProbabilityTheory.Kernel.Indep m₁ m₂ κ μ) {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ProbabilityTheory.Kernel.IndepSet s t κ μ
-- ProbabilityTheory.Kernel.indep_iff_forall_indepSet
theorem indep_iff_forall_indepSet (m₁ m₂ : MeasurableSpace Ω) {_m0 : MeasurableSpace Ω}
    (κ : Kernel α Ω) (μ : Measure α) :
    Indep m₁ m₂ κ μ ↔ ∀ s t, MeasurableSet[m₁] s → MeasurableSet[m₂] t → IndepSet s t κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} (m₁ m₂ : MeasurableSpace Ω) {_m0 : MeasurableSpace Ω} (κ : ProbabilityTheory.Kernel α Ω) (μ : MeasureTheory.Measure α)
Goal: ProbabilityTheory.Kernel.Indep m₁ m₂ κ μ ↔
  ∀ (s t : Set Ω), MeasurableSet s → MeasurableSet t → ProbabilityTheory.Kernel.IndepSet s t κ μ
-- ProbabilityTheory.Kernel.indepFun_iff_measure_inter_preimage_eq_mul
theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
    {mβ' : MeasurableSpace β'} :
    IndepFun f g κ μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t
        → ∀ᵐ a ∂μ, κ a (f ⁻¹' s ∩ g ⁻¹' t) = κ a (f ⁻¹' s) * κ a (g ⁻¹' t) 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
Goal: ProbabilityTheory.Kernel.IndepFun f g κ μ ↔
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s → MeasurableSet t → ∀ᵐ (a : α) ∂μ, (κ a) (f ⁻¹' s ∩ g ⁻¹' t) = (κ a) (f ⁻¹' s) * (κ a) (g ⁻¹' t)
-- ProbabilityTheory.Kernel.iIndepFun_iff_measure_inter_preimage_eq_mul
theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*}
    (m : ∀ x, MeasurableSpace (β x)) (f : ∀ i, Ω → β i) :
    iIndepFun m f κ μ ↔
      ∀ (S : Finset ι) {sets : ∀ i : ι, Set (β i)} (_H : ∀ i, i ∈ S → MeasurableSet[m i] (sets i)),
        ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, (f i) ⁻¹' (sets i)) = ∏ i ∈ S, κ a ((f i) ⁻¹' (sets i)) 
Arguments: {α : Type u_1} {Ω : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {ι : Type u_8} {β : ι → Type u_9} (m : (x : ι) → MeasurableSpace (β x)) (f : (i : ι) → Ω → β i)
Goal: ProbabilityTheory.Kernel.iIndepFun m f κ μ ↔
  ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},
    (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, f i ⁻¹' sets i) = ∏ i ∈ S, (κ a) (f i ⁻¹' sets i)
-- ProbabilityTheory.Kernel.indepFun_iff_indepSet_preimage
theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    [IsMarkovKernel κ] (hf : Measurable f) (hg : Measurable g) :
    IndepFun f g κ μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} [ProbabilityTheory.IsMarkovKernel κ] (hf : Measurable f) (hg : Measurable g)
Goal: ProbabilityTheory.Kernel.IndepFun f g κ μ ↔
  ∀ (s : Set β) (t : Set β'),
    MeasurableSet s → MeasurableSet t → ProbabilityTheory.Kernel.IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ
-- ProbabilityTheory.Kernel.IndepFun.symm
@[symm]
nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {_ : MeasurableSpace β'}
    (hfg : IndepFun f g κ μ) : IndepFun g f κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'}
Goal: ∀ {x : MeasurableSpace β} {x_1 : MeasurableSpace β'},
  ProbabilityTheory.Kernel.IndepFun f g κ μ → ProbabilityTheory.Kernel.IndepFun g f κ μ
-- ProbabilityTheory.Kernel.IndepFun.ae_eq
theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    {f' : Ω → β} {g' : Ω → β'} (hfg : IndepFun f g κ μ)
    (hf : ∀ᵐ a ∂μ, f =ᵐ[κ a] f') (hg : ∀ᵐ a ∂μ, g =ᵐ[κ a] g') :
    IndepFun f' g' κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f' : Ω → β} {g' : Ω → β'} (hfg : ProbabilityTheory.Kernel.IndepFun f g κ μ) (hf : ∀ᵐ (a : α) ∂μ, f =ᵐ[κ a] f') (hg : ∀ᵐ (a : α) ∂μ, g =ᵐ[κ a] g')
Goal: ProbabilityTheory.Kernel.IndepFun f' g' κ μ
-- ProbabilityTheory.Kernel.IndepFun.comp
theorem IndepFun.comp {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    {mγ : MeasurableSpace γ} {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'}
    (hfg : IndepFun f g κ μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
    IndepFun (φ ∘ f) (ψ ∘ g) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {γ : Type u_6} {γ' : Type u_7} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {mγ : MeasurableSpace γ} {mγ' : MeasurableSpace γ'} {φ : β → γ} {ψ : β' → γ'} (hfg : ProbabilityTheory.Kernel.IndepFun f g κ μ) (hφ : Measurable φ) (hψ : Measurable ψ)
Goal: ProbabilityTheory.Kernel.IndepFun (φ ∘ f) (ψ ∘ g) κ μ
-- ProbabilityTheory.Kernel.IndepFun.neg_right
theorem IndepFun.neg_right {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β']
    [MeasurableNeg β'] (hfg : IndepFun f g κ μ)  :
    IndepFun f (-g) κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β'] [MeasurableNeg β'] (hfg : ProbabilityTheory.Kernel.IndepFun f g κ μ)
Goal: ProbabilityTheory.Kernel.IndepFun f (-g) κ μ
-- ProbabilityTheory.Kernel.IndepFun.neg_left
theorem IndepFun.neg_left {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β]
    [MeasurableNeg β] (hfg : IndepFun f g κ μ) :
    IndepFun (-f) g κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {β : Type u_4} {β' : Type u_5} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β] [MeasurableNeg β] (hfg : ProbabilityTheory.Kernel.IndepFun f g κ μ)
Goal: ProbabilityTheory.Kernel.IndepFun (-f) g κ μ
-- ProbabilityTheory.Kernel.iIndepFun.of_subsingleton
@[nontriviality]
lemma iIndepFun.of_subsingleton [IsMarkovKernel κ] [Subsingleton ι] : iIndepFun m f κ μ 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} [ProbabilityTheory.IsMarkovKernel κ] [Subsingleton ι]
Goal: ProbabilityTheory.Kernel.iIndepFun m f κ μ
-- ProbabilityTheory.Kernel.iIndepFun.ae_isProbabilityMeasure
lemma iIndepFun.ae_isProbabilityMeasure (h : iIndepFun m f κ μ) :
    ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a) 
Arguments: {α : Type u_1} {Ω : Type u_2} {ι : Type u_3} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ProbabilityTheory.Kernel α Ω} {μ : MeasureTheory.Measure α} {β : ι → Type u_8} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} (h : ProbabilityTheory.Kernel.iIndepFun m f κ μ)
Goal: ∀ᵐ (a : α) ∂μ, MeasureTheory.IsProbabilityMeasure (κ a)