ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.IndepSets.union
theorem IndepSets.union {s₁ s₂ s' : Set (Set Ω)} (h₁ : IndepSets s₁ s' μ) (h₂ : IndepSets s₂ s' μ) :
    IndepSets (s₁ ∪ s₂) s' μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s₁ s₂ s' : Set (Set Ω)} (h₁ : ProbabilityTheory.IndepSets s₁ s' μ) (h₂ : ProbabilityTheory.IndepSets s₂ s' μ)
Goal: ProbabilityTheory.IndepSets (s₁ ∪ s₂) s' μ
-- ProbabilityTheory.IndepSets.union_iff
@[simp]
theorem IndepSets.union_iff {s₁ s₂ s' : Set (Set Ω)} :
    IndepSets (s₁ ∪ s₂) s' μ ↔ IndepSets s₁ s' μ ∧ IndepSets s₂ s' μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s₁ s₂ s' : Set (Set Ω)}
Goal: ProbabilityTheory.IndepSets (s₁ ∪ s₂) s' μ ↔ ProbabilityTheory.IndepSets s₁ s' μ ∧ ProbabilityTheory.IndepSets s₂ s' μ
-- ProbabilityTheory.IndepSets.iUnion
theorem IndepSets.iUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    (hyp : ∀ n, IndepSets (s n) s' μ) :
    IndepSets (⋃ n, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} (hyp : ∀ (n : ι), ProbabilityTheory.IndepSets (s n) s' μ)
Goal: ProbabilityTheory.IndepSets (⋃ n, s n) s' μ
-- ProbabilityTheory.IndepSets.bUnion
theorem IndepSets.bUnion {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    {u : Set ι} (hyp : ∀ n ∈ u, IndepSets (s n) s' μ) :
    IndepSets (⋃ n ∈ u, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {u : Set ι} (hyp : ∀ n ∈ u, ProbabilityTheory.IndepSets (s n) s' μ)
Goal: ProbabilityTheory.IndepSets (⋃ n ∈ u, s n) s' μ
-- ProbabilityTheory.IndepSets.inter
theorem IndepSets.inter {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) (h₁ : IndepSets s₁ s' μ) :
    IndepSets (s₁ ∩ s₂) s' μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s₁ s' : Set (Set Ω)} (s₂ : Set (Set Ω)) (h₁ : ProbabilityTheory.IndepSets s₁ s' μ)
Goal: ProbabilityTheory.IndepSets (s₁ ∩ s₂) s' μ
-- ProbabilityTheory.IndepSets.iInter
theorem IndepSets.iInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    (h : ∃ n, IndepSets (s n) s' μ) :
    IndepSets (⋂ n, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} (h : ∃ n, ProbabilityTheory.IndepSets (s n) s' μ)
Goal: ProbabilityTheory.IndepSets (⋂ n, s n) s' μ
-- ProbabilityTheory.IndepSets.bInter
theorem IndepSets.bInter {s : ι → Set (Set Ω)} {s' : Set (Set Ω)}
    {u : Set ι} (h : ∃ n ∈ u, IndepSets (s n) s' μ) :
    IndepSets (⋂ n ∈ u, s n) s' μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} {s' : Set (Set Ω)} {u : Set ι} (h : ∃ n ∈ u, ProbabilityTheory.IndepSets (s n) s' μ)
Goal: ProbabilityTheory.IndepSets (⋂ n ∈ u, s n) s' μ
-- ProbabilityTheory.indepSets_singleton_iff
theorem indepSets_singleton_iff {s t : Set Ω} :
    IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s t : Set Ω}
Goal: ProbabilityTheory.IndepSets {s} {t} μ ↔ μ (s ∩ t) = μ s * μ t
-- ProbabilityTheory.iIndepSets.indepSets
theorem iIndepSets.indepSets {s : ι → Set (Set Ω)}
    (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} (h_indep : ProbabilityTheory.iIndepSets s μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.IndepSets (s i) (s j) μ
-- ProbabilityTheory.iIndep.indep
theorem iIndep.indep
    (h_indep : iIndep m μ) {i j : ι} (hij : i ≠ j) : Indep (m i) (m j) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (h_indep : ProbabilityTheory.iIndep m μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.Indep (m i) (m j) μ
-- ProbabilityTheory.iIndepFun.indepFun
theorem iIndepFun.indepFun {β : ι → Type*}
    {m : ∀ x, MeasurableSpace (β x)} {f : ∀ i, Ω → β i} (hf_Indep : iIndepFun m f μ) {i j : ι}
    (hij : i ≠ j) :
    IndepFun (f i) (f j) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {β : ι → Type u_6} {m : (x : ι) → MeasurableSpace (β x)} {f : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.iIndepFun m f μ) {i j : ι} (hij : i ≠ j)
Goal: ProbabilityTheory.IndepFun (f i) (f j) μ
-- ProbabilityTheory.iIndep.iIndepSets
theorem iIndep.iIndepSets
    {s : ι → Set (Set Ω)} (hms : ∀ n, m n = generateFrom (s n)) (h_indep : iIndep m μ) :
    iIndepSets s μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set (Set Ω)} (hms : ∀ (n : ι), m n = MeasurableSpace.generateFrom (s n)) (h_indep : ProbabilityTheory.iIndep m μ)
Goal: ProbabilityTheory.iIndepSets s μ
-- ProbabilityTheory.Indep.indepSets
theorem Indep.indepSets {s1 s2 : Set (Set Ω)}
    (h_indep : Indep (generateFrom s1) (generateFrom s2) μ) :
    IndepSets s1 s2 μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s1 s2 : Set (Set Ω)} (h_indep : ProbabilityTheory.Indep (MeasurableSpace.generateFrom s1) (MeasurableSpace.generateFrom s2) μ)
Goal: ProbabilityTheory.IndepSets s1 s2 μ
-- ProbabilityTheory.IndepSets.indep
theorem IndepSets.indep [IsProbabilityMeasure μ]
    {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 : IndepSets p1 p2 μ) :
    Indep m1 m2 μ 
Arguments: {Ω : Type u_1} {m1 m2 _mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure 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.IndepSets p1 p2 μ)
Goal: ProbabilityTheory.Indep m1 m2 μ
-- ProbabilityTheory.IndepSets.indep'
theorem IndepSets.indep' [IsProbabilityMeasure μ]
    {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 μ) :
    Indep (generateFrom p1) (generateFrom p2) μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s) (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : ProbabilityTheory.IndepSets p1 p2 μ)
Goal: ProbabilityTheory.Indep (MeasurableSpace.generateFrom p1) (MeasurableSpace.generateFrom p2) μ
-- ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
    {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
    IndepSets (piiUnionInter s S) (piiUnionInter s T) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {s : ι → Set (Set Ω)} {S T : Set ι} (h_indep : ProbabilityTheory.iIndepSets s μ) (hST : Disjoint S T)
Goal: ProbabilityTheory.IndepSets (piiUnionInter s S) (piiUnionInter s T) μ
-- ProbabilityTheory.iIndepSet.indep_generateFrom_of_disjoint
theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
    Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T)
Goal: ProbabilityTheory.Indep (MeasurableSpace.generateFrom {t | ∃ n ∈ S, s n = t})
  (MeasurableSpace.generateFrom {t | ∃ k ∈ T, s k = t}) μ
-- ProbabilityTheory.indep_iSup_of_disjoint
theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ]
    (h_le : ∀ i, m i ≤ _mΩ) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
    Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h_le : ∀ (i : ι), m i ≤ _mΩ) (h_indep : ProbabilityTheory.iIndep m μ) {S T : Set ι} (hST : Disjoint S T)
Goal: ProbabilityTheory.Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ
-- ProbabilityTheory.indep_iSup_of_directed_le
theorem indep_iSup_of_directed_le
    [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m1 μ)
    (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Directed (· ≤ ·) m) :
    Indep (⨆ i, m i) m1 μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {m1 _mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h_indep : ∀ (i : ι), ProbabilityTheory.Indep (m i) m1 μ) (h_le : ∀ (i : ι), m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Directed (fun x x_1 => x ≤ x_1) m)
Goal: ProbabilityTheory.Indep (⨆ i, m i) m1 μ
-- ProbabilityTheory.iIndepSet.indep_generateFrom_lt
theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
    Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [Preorder ι] [MeasureTheory.IsProbabilityMeasure μ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (i : ι)
Goal: ProbabilityTheory.Indep (MeasurableSpace.generateFrom {s i}) (MeasurableSpace.generateFrom {t | ∃ j < i, s j = t}) μ
-- ProbabilityTheory.iIndepSet.indep_generateFrom_le
theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
    Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [LinearOrder ι] [MeasureTheory.IsProbabilityMeasure μ] {s : ι → Set Ω} (hsm : ∀ (n : ι), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (i : ι) {k : ι} (hk : i < k)
Goal: ProbabilityTheory.Indep (MeasurableSpace.generateFrom {s k}) (MeasurableSpace.generateFrom {t | ∃ j ≤ i, s j = t}) μ
-- ProbabilityTheory.iIndepSet.indep_generateFrom_le_nat
theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
    (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
    Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {s : ℕ → Set Ω} (hsm : ∀ (n : ℕ), MeasurableSet (s n)) (hs : ProbabilityTheory.iIndepSet s μ) (n : ℕ)
Goal: ProbabilityTheory.Indep (MeasurableSpace.generateFrom {s (n + 1)}) (MeasurableSpace.generateFrom {t | ∃ k ≤ n, s k = t})
  μ
-- ProbabilityTheory.indep_iSup_of_monotone
theorem indep_iSup_of_monotone [SemilatticeSup ι] [IsProbabilityMeasure μ]
    (h_indep : ∀ i, Indep (m i) m1 μ) (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Monotone m) :
    Indep (⨆ i, m i) m1 μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {m1 _mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeSup ι] [MeasureTheory.IsProbabilityMeasure μ] (h_indep : ∀ (i : ι), ProbabilityTheory.Indep (m i) m1 μ) (h_le : ∀ (i : ι), m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Monotone m)
Goal: ProbabilityTheory.Indep (⨆ i, m i) m1 μ
-- ProbabilityTheory.indep_iSup_of_antitone
theorem indep_iSup_of_antitone [SemilatticeInf ι] [IsProbabilityMeasure μ]
    (h_indep : ∀ i, Indep (m i) m1 μ) (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Antitone m) :
    Indep (⨆ i, m i) m1 μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {m1 _mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [SemilatticeInf ι] [MeasureTheory.IsProbabilityMeasure μ] (h_indep : ∀ (i : ι), ProbabilityTheory.Indep (m i) m1 μ) (h_le : ∀ (i : ι), m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Antitone m)
Goal: ProbabilityTheory.Indep (⨆ i, m i) m1 μ
-- ProbabilityTheory.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 μ) (haS : a ∉ S) :
    IndepSets (piiUnionInter π S) (π a) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {π : ι → Set (Set Ω)} {a : ι} {S : Finset ι} (hp_ind : ProbabilityTheory.iIndepSets π μ) (haS : a ∉ S)
Goal: ProbabilityTheory.IndepSets (piiUnionInter π ↑S) (π a) μ
-- ProbabilityTheory.iIndepSets.iIndep
theorem iIndepSets.iIndep [IsProbabilityMeasure μ]
    (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 μ) :
    iIndep m μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (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.iIndepSets π μ)
Goal: ProbabilityTheory.iIndep m μ
-- ProbabilityTheory.indepSet_iff_indepSets_singleton
theorem indepSet_iff_indepSets_singleton (hs_meas : MeasurableSet s)
    (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {s t : Set Ω} (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (μ : MeasureTheory.Measure Ω := by volume_tac) [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.IndepSet s t μ ↔ ProbabilityTheory.IndepSets {s} {t} μ
-- ProbabilityTheory.indepSet_iff_measure_inter_eq_mul
theorem indepSet_iff_measure_inter_eq_mul (hs_meas : MeasurableSet s)
    (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {s t : Set Ω} (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (μ : MeasureTheory.Measure Ω := by volume_tac) [MeasureTheory.IsProbabilityMeasure μ]
Goal: ProbabilityTheory.IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t
-- ProbabilityTheory.IndepSets.indepSet_of_mem
theorem IndepSets.indepSet_of_mem (hs : s ∈ S) (ht : t ∈ T)
    (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
    (μ : Measure Ω := by volume_tac) [IsProbabilityMeasure μ]
    (h_indep : IndepSets S T μ) :
    IndepSet s t μ 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {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 := by volume_tac) [MeasureTheory.IsProbabilityMeasure μ] (h_indep : ProbabilityTheory.IndepSets S T μ)
Goal: ProbabilityTheory.IndepSet s t μ
-- ProbabilityTheory.Indep.indepSet_of_measurableSet
theorem Indep.indepSet_of_measurableSet
    (h_indep : Indep m₁ m₂ μ) {s t : Set Ω} (hs : MeasurableSet[m₁] s) (ht : MeasurableSet[m₂] t) :
    IndepSet s t μ 
Arguments: {Ω : Type u_1} {m₁ m₂ _mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} (h_indep : ProbabilityTheory.Indep m₁ m₂ μ) {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ProbabilityTheory.IndepSet s t μ
-- ProbabilityTheory.indep_iff_forall_indepSet
theorem indep_iff_forall_indepSet (μ : Measure Ω) :
    Indep m₁ m₂ μ ↔ ∀ s t, MeasurableSet[m₁] s → MeasurableSet[m₂] t → IndepSet s t μ 
Arguments: {Ω : Type u_1} {m₁ m₂ _mΩ : MeasurableSpace Ω} (μ : MeasureTheory.Measure Ω)
Goal: ProbabilityTheory.Indep m₁ m₂ μ ↔ ∀ (s t : Set Ω), MeasurableSet s → MeasurableSet t → ProbabilityTheory.IndepSet s t μ
-- ProbabilityTheory.iIndep_comap_mem_iff
theorem iIndep_comap_mem_iff {f : ι → Set Ω} :
    iIndep (fun i => MeasurableSpace.comap (· ∈ f i) ⊤) μ ↔ iIndepSet f μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ι → Set Ω}
Goal: ProbabilityTheory.iIndep (fun i => MeasurableSpace.comap (fun x => x ∈ f i) ⊤) μ ↔ ProbabilityTheory.iIndepSet f μ
-- ProbabilityTheory.iIndepSets_singleton_iff
theorem iIndepSets_singleton_iff {s : ι → Set Ω} :
    iIndepSets (fun i ↦ {s i}) μ ↔ ∀ t, μ (⋂ i ∈ t, s i) = ∏ i ∈ t, μ (s i) 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {s : ι → Set Ω}
Goal: ProbabilityTheory.iIndepSets (fun i => {s i}) μ ↔ ∀ (t : Finset ι), μ (⋂ i ∈ t, s i) = ∏ i ∈ t, μ (s i)
-- ProbabilityTheory.iIndepSet_iff_iIndepSets_singleton
theorem iIndepSet_iff_iIndepSets_singleton {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
    iIndepSet f μ ↔ iIndepSets (fun i ↦ {f i}) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {f : ι → Set Ω} (hf : ∀ (i : ι), MeasurableSet (f i))
Goal: ProbabilityTheory.iIndepSet f μ ↔ ProbabilityTheory.iIndepSets (fun i => {f i}) μ
-- ProbabilityTheory.iIndepSet_iff_meas_biInter
theorem iIndepSet_iff_meas_biInter {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
    iIndepSet f μ ↔ ∀ s, μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {f : ι → Set Ω} (hf : ∀ (i : ι), MeasurableSet (f i))
Goal: ProbabilityTheory.iIndepSet f μ ↔ ∀ (s : Finset ι), μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)
-- ProbabilityTheory.iIndepSets.iIndepSet_of_mem
theorem iIndepSets.iIndepSet_of_mem {π : ι → Set (Set Ω)} {f : ι → Set Ω}
    (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
    (hπ : iIndepSets π μ) : iIndepSet f μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {π : ι → Set (Set Ω)} {f : ι → Set Ω} (hfπ : ∀ (i : ι), f i ∈ π i) (hf : ∀ (i : ι), MeasurableSet (f i)) (hπ : ProbabilityTheory.iIndepSets π μ)
Goal: ProbabilityTheory.iIndepSet f μ
-- ProbabilityTheory.indepFun_iff_measure_inter_preimage_eq_mul
theorem indepFun_iff_measure_inter_preimage_eq_mul {mβ : MeasurableSpace β}
    {mβ' : MeasurableSpace β'} :
    IndepFun f g μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t
        → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t) 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
Goal: ProbabilityTheory.IndepFun f g μ ↔
  ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → μ (f ⁻¹' s ∩ g ⁻¹' t) = μ (f ⁻¹' s) * μ (g ⁻¹' t)
-- ProbabilityTheory.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 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) = ∏ i ∈ S, μ (f i ⁻¹' sets i) 
Arguments: {Ω : Type u_1} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_10} {β : ι → Type u_11} (m : (x : ι) → MeasurableSpace (β x)) (f : (i : ι) → Ω → β i)
Goal: ProbabilityTheory.iIndepFun 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) = ∏ i ∈ S, μ (f i ⁻¹' sets i)
-- ProbabilityTheory.indepFun_iff_indepSet_preimage
theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
    IndepFun f g μ ↔
      ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} [MeasureTheory.IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g)
Goal: ProbabilityTheory.IndepFun f g μ ↔
  ∀ (s : Set β) (t : Set β'), MeasurableSet s → MeasurableSet t → ProbabilityTheory.IndepSet (f ⁻¹' s) (g ⁻¹' t) μ
-- ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map
theorem indepFun_iff_map_prod_eq_prod_map_map {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    [IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
    IndepFun f g μ ↔ μ.map (fun ω ↦ (f ω, g ω)) = (μ.map f).prod (μ.map g) 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} [MeasureTheory.IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
Goal: ProbabilityTheory.IndepFun f g μ ↔
  MeasureTheory.Measure.map (fun ω => (f ω, g ω)) μ =
    (MeasureTheory.Measure.map f μ).prod (MeasureTheory.Measure.map g μ)
-- ProbabilityTheory.IndepFun.symm
@[symm]
nonrec theorem IndepFun.symm {_ : MeasurableSpace β} {_ : MeasurableSpace β'}
    (hfg : IndepFun f g μ) : IndepFun g f μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'}
Goal: ∀ {x : MeasurableSpace β} {x_1 : MeasurableSpace β'},
  ProbabilityTheory.IndepFun f g μ → ProbabilityTheory.IndepFun g f μ
-- ProbabilityTheory.IndepFun.ae_eq
theorem IndepFun.ae_eq {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
    {f' : Ω → β} {g' : Ω → β'} (hfg : IndepFun f g μ)
    (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g') : IndepFun f' g' μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f' : Ω → β} {g' : Ω → β'} (hfg : ProbabilityTheory.IndepFun f g μ) (hf : f =ᵐ[μ] f') (hg : g =ᵐ[μ] g')
Goal: ProbabilityTheory.IndepFun f' g' μ
-- ProbabilityTheory.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 μ) (hφ : Measurable φ) (hψ : Measurable ψ) :
    IndepFun (φ ∘ f) (ψ ∘ g) μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {γ : Type u_8} {γ' : Type u_9} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {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.IndepFun f g μ) (hφ : Measurable φ) (hψ : Measurable ψ)
Goal: ProbabilityTheory.IndepFun (φ ∘ f) (ψ ∘ g) μ
-- ProbabilityTheory.IndepFun.neg_right
theorem IndepFun.neg_right {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β']
    [MeasurableNeg β'] (hfg : IndepFun f g μ) :
    IndepFun f (-g) μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β'] [MeasurableNeg β'] (hfg : ProbabilityTheory.IndepFun f g μ)
Goal: ProbabilityTheory.IndepFun f (-g) μ
-- ProbabilityTheory.IndepFun.neg_left
theorem IndepFun.neg_left {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β]
    [MeasurableNeg β] (hfg : IndepFun f g μ) :
    IndepFun (-f) g μ 
Arguments: {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {_mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β] [MeasurableNeg β] (hfg : ProbabilityTheory.IndepFun f g μ)
Goal: ProbabilityTheory.IndepFun (-f) g μ