ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Turing.PartrecToTM2.codeSupp_comp
@[simp]
theorem codeSupp_comp (f g k) :
    codeSupp (Code.comp f g) k =
      trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) 
Arguments: (f g : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.codeSupp (f.comp g) k =
  Turing.PartrecToTM2.trStmts₁ (Turing.PartrecToTM2.trNormal (f.comp g) k) ∪
    Turing.PartrecToTM2.codeSupp g (Turing.PartrecToTM2.Cont'.comp f k)
-- Turing.PartrecToTM2.codeSupp_case
@[simp]
theorem codeSupp_case (f g k) :
    codeSupp (Code.case f g) k =
      trStmts₁ (trNormal (Code.case f g) k) ∪ (codeSupp f k ∪ codeSupp g k) 
Arguments: (f g : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.codeSupp (f.case g) k =
  Turing.PartrecToTM2.trStmts₁ (Turing.PartrecToTM2.trNormal (f.case g) k) ∪
    (Turing.PartrecToTM2.codeSupp f k ∪ Turing.PartrecToTM2.codeSupp g k)
-- Turing.PartrecToTM2.codeSupp_fix
@[simp]
theorem codeSupp_fix (f k) :
    codeSupp (Code.fix f) k = trStmts₁ (trNormal (Code.fix f) k) ∪ codeSupp f (Cont'.fix f k) 
Arguments: (f : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.codeSupp f.fix k =
  Turing.PartrecToTM2.trStmts₁ (Turing.PartrecToTM2.trNormal f.fix k) ∪
    Turing.PartrecToTM2.codeSupp f (Turing.PartrecToTM2.Cont'.fix f k)
-- Turing.PartrecToTM2.contSupp_cons₁
@[simp]
theorem contSupp_cons₁ (fs k) :
    contSupp (Cont'.cons₁ fs k) =
      trStmts₁
          (move₂ (fun _ => false) main aux <|
            move₂ (fun s => s = Γ'.consₗ) stack main <|
              move₂ (fun _ => false) aux stack <| trNormal fs (Cont'.cons₂ k)) ∪
        codeSupp fs (Cont'.cons₂ k) 
Arguments: (fs : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.contSupp (Turing.PartrecToTM2.Cont'.cons₁ fs k) =
  Turing.PartrecToTM2.trStmts₁
      (Turing.PartrecToTM2.move₂ (fun x => false) Turing.PartrecToTM2.K'.main Turing.PartrecToTM2.K'.aux
        (Turing.PartrecToTM2.move₂ (fun s => decide (s = Turing.PartrecToTM2.Γ'.consₗ)) Turing.PartrecToTM2.K'.stack
          Turing.PartrecToTM2.K'.main
          (Turing.PartrecToTM2.move₂ (fun x => false) Turing.PartrecToTM2.K'.aux Turing.PartrecToTM2.K'.stack
            (Turing.PartrecToTM2.trNormal fs k.cons₂)))) ∪
    Turing.PartrecToTM2.codeSupp fs k.cons₂
-- Turing.PartrecToTM2.contSupp_cons₂
@[simp]
theorem contSupp_cons₂ (k) :
    contSupp (Cont'.cons₂ k) = trStmts₁ (head stack <| Λ'.ret k) ∪ contSupp k 
Arguments: (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.contSupp k.cons₂ =
  Turing.PartrecToTM2.trStmts₁ (Turing.PartrecToTM2.head Turing.PartrecToTM2.K'.stack (Turing.PartrecToTM2.Λ'.ret k)) ∪
    Turing.PartrecToTM2.contSupp k
-- Turing.PartrecToTM2.contSupp_comp
@[simp]
theorem contSupp_comp (f k) : contSupp (Cont'.comp f k) = codeSupp f k 
Arguments: (f : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.contSupp (Turing.PartrecToTM2.Cont'.comp f k) = Turing.PartrecToTM2.codeSupp f k
-- Turing.PartrecToTM2.contSupp_fix
theorem contSupp_fix (f k) : contSupp (Cont'.fix f k) = codeSupp f (Cont'.fix f k) 
Arguments: (f : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.PartrecToTM2.contSupp (Turing.PartrecToTM2.Cont'.fix f k) =
  Turing.PartrecToTM2.codeSupp f (Turing.PartrecToTM2.Cont'.fix f k)
-- Turing.PartrecToTM2.contSupp_halt
@[simp]
theorem contSupp_halt : contSupp Cont'.halt = ∅ 
Goal: Turing.PartrecToTM2.contSupp Turing.PartrecToTM2.Cont'.halt = ∅
-- Turing.PartrecToTM2.supports_insert
theorem supports_insert {K S q} :
    Supports (insert q K) S ↔ TM2.SupportsStmt S (tr q) ∧ Supports K S 
Arguments: {K S : Finset Turing.PartrecToTM2.Λ'} {q : Turing.PartrecToTM2.Λ'}
Goal: Turing.PartrecToTM2.Supports (insert q K) S ↔
  Turing.TM2.SupportsStmt S (Turing.PartrecToTM2.tr q) ∧ Turing.PartrecToTM2.Supports K S
-- Turing.PartrecToTM2.supports_singleton
theorem supports_singleton {S q} : Supports {q} S ↔ TM2.SupportsStmt S (tr q) 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {q : Turing.PartrecToTM2.Λ'}
Goal: Turing.PartrecToTM2.Supports {q} S ↔ Turing.TM2.SupportsStmt S (Turing.PartrecToTM2.tr q)
-- Turing.PartrecToTM2.supports_union
theorem supports_union {K₁ K₂ S} : Supports (K₁ ∪ K₂) S ↔ Supports K₁ S ∧ Supports K₂ S 
Arguments: {K₁ K₂ S : Finset Turing.PartrecToTM2.Λ'}
Goal: Turing.PartrecToTM2.Supports (K₁ ∪ K₂) S ↔ Turing.PartrecToTM2.Supports K₁ S ∧ Turing.PartrecToTM2.Supports K₂ S
-- Turing.PartrecToTM2.supports_biUnion
theorem supports_biUnion {K : Option Γ' → Finset Λ'} {S} :
    Supports (Finset.univ.biUnion K) S ↔ ∀ a, Supports (K a) S 
Arguments: {K : Option Turing.PartrecToTM2.Γ' → Finset Turing.PartrecToTM2.Λ'} {S : Finset Turing.PartrecToTM2.Λ'}
Goal: Turing.PartrecToTM2.Supports (Finset.univ.biUnion K) S ↔
  ∀ (a : Option Turing.PartrecToTM2.Γ'), Turing.PartrecToTM2.Supports (K a) S
-- Turing.PartrecToTM2.head_supports
theorem head_supports {S k q} (H : (q : Λ').Supports S) : (head k q).Supports S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {k : Turing.PartrecToTM2.K'} {q : Turing.PartrecToTM2.Λ'} (H : Turing.PartrecToTM2.Λ'.Supports S q)
Goal: Turing.PartrecToTM2.Λ'.Supports S (Turing.PartrecToTM2.head k q)
-- Turing.PartrecToTM2.ret_supports
theorem ret_supports {S k} (H₁ : contSupp k ⊆ S) : TM2.SupportsStmt S (tr (Λ'.ret k)) 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {k : Turing.PartrecToTM2.Cont'} (H₁ : Turing.PartrecToTM2.contSupp k ⊆ S)
Goal: Turing.TM2.SupportsStmt S (Turing.PartrecToTM2.tr (Turing.PartrecToTM2.Λ'.ret k))
-- Turing.PartrecToTM2.trStmts₁_supports
theorem trStmts₁_supports {S q} (H₁ : (q : Λ').Supports S) (HS₁ : trStmts₁ q ⊆ S) :
    Supports (trStmts₁ q) S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {q : Turing.PartrecToTM2.Λ'} (H₁ : Turing.PartrecToTM2.Λ'.Supports S q) (HS₁ : Turing.PartrecToTM2.trStmts₁ q ⊆ S)
Goal: Turing.PartrecToTM2.Supports (Turing.PartrecToTM2.trStmts₁ q) S
-- Turing.PartrecToTM2.trStmts₁_supports'
theorem trStmts₁_supports' {S q K} (H₁ : (q : Λ').Supports S) (H₂ : trStmts₁ q ∪ K ⊆ S)
    (H₃ : K ⊆ S → Supports K S) : Supports (trStmts₁ q ∪ K) S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {q : Turing.PartrecToTM2.Λ'} {K : Finset Turing.PartrecToTM2.Λ'} (H₁ : Turing.PartrecToTM2.Λ'.Supports S q) (H₂ : Turing.PartrecToTM2.trStmts₁ q ∪ K ⊆ S) (H₃ : K ⊆ S → Turing.PartrecToTM2.Supports K S)
Goal: Turing.PartrecToTM2.Supports (Turing.PartrecToTM2.trStmts₁ q ∪ K) S
-- Turing.PartrecToTM2.trNormal_supports
theorem trNormal_supports {S c k} (Hk : codeSupp c k ⊆ S) : (trNormal c k).Supports S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {c : Turing.ToPartrec.Code} {k : Turing.PartrecToTM2.Cont'} (Hk : Turing.PartrecToTM2.codeSupp c k ⊆ S)
Goal: Turing.PartrecToTM2.Λ'.Supports S (Turing.PartrecToTM2.trNormal c k)
-- Turing.PartrecToTM2.codeSupp'_supports
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {c : Turing.ToPartrec.Code} {k : Turing.PartrecToTM2.Cont'} (H : Turing.PartrecToTM2.codeSupp c k ⊆ S)
Goal: Turing.PartrecToTM2.Supports (Turing.PartrecToTM2.codeSupp' c k) S
-- Turing.PartrecToTM2.contSupp_supports
theorem contSupp_supports {S k} (H : contSupp k ⊆ S) : Supports (contSupp k) S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {k : Turing.PartrecToTM2.Cont'} (H : Turing.PartrecToTM2.contSupp k ⊆ S)
Goal: Turing.PartrecToTM2.Supports (Turing.PartrecToTM2.contSupp k) S
-- Turing.PartrecToTM2.codeSupp_supports
theorem codeSupp_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp c k) S 
Arguments: {S : Finset Turing.PartrecToTM2.Λ'} {c : Turing.ToPartrec.Code} {k : Turing.PartrecToTM2.Cont'} (H : Turing.PartrecToTM2.codeSupp c k ⊆ S)
Goal: Turing.PartrecToTM2.Supports (Turing.PartrecToTM2.codeSupp c k) S
-- Turing.PartrecToTM2.tr_supports
theorem tr_supports (c k) : @TM2.Supports _ _ _ _ ⟨trNormal c k⟩ tr (codeSupp c k) 
Arguments: (c : Turing.ToPartrec.Code) (k : Turing.PartrecToTM2.Cont')
Goal: Turing.TM2.Supports Turing.PartrecToTM2.tr (Turing.PartrecToTM2.codeSupp c k)
-- ProbabilityTheory.iCondIndepSets_iff
lemma iCondIndepSets_iff (π : ι → Set (Set Ω)) (hπ : ∀ i s (_hs : s ∈ π i), MeasurableSet s)
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepSets m' hm' π μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → f i ∈ π i),
      μ⟦⋂ i ∈ s, f i | m'⟧ =ᵐ[μ] ∏ i ∈ s, (μ⟦f i | m'⟧) 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (π : ι → Set (Set Ω)) (hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepSets m' hm' π μ ↔
  ∀ (s : Finset ι) {f : ι → Set Ω},
    (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndepSets_iff
lemma condIndepSets_iff (s1 s2 : Set (Set Ω)) (hs1 : ∀ s ∈ s1, MeasurableSet s)
    (hs2 : ∀ s ∈ s2, MeasurableSet s) (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepSets m' hm' s1 s2 μ ↔ ∀ (t1 t2 : Set Ω) (_ : t1 ∈ s1) (_ : t2 ∈ s2),
      (μ⟦t1 ∩ t2 | m'⟧) =ᵐ[μ] (μ⟦t1 | m'⟧) * (μ⟦t2 | m'⟧) 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s1 s2 : Set (Set Ω)) (hs1 : ∀ s ∈ s1, MeasurableSet s) (hs2 : ∀ s ∈ s2, MeasurableSet s) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepSets m' hm' s1 s2 μ ↔
  ∀ (t1 t2 : Set Ω),
    t1 ∈ s1 →
      t2 ∈ s2 → μ[(t1 ∩ t2).indicator fun ω => 1|m'] =ᵐ[μ] μ[t1.indicator fun ω => 1|m'] * μ[t2.indicator fun ω => 1|m']
-- ProbabilityTheory.iCondIndepSets_singleton_iff
lemma iCondIndepSets_singleton_iff (s : ι → Set Ω) (hπ : ∀ i, MeasurableSet (s i))
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepSets m' hm' (fun i ↦ {s i}) μ ↔ ∀ S : Finset ι,
      μ⟦⋂ i ∈ S, s i | m'⟧ =ᵐ[μ] ∏ i ∈ S, (μ⟦s i | m'⟧) 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s : ι → Set Ω) (hπ : ∀ (i : ι), MeasurableSet (s i)) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepSets m' hm' (fun i => {s i}) μ ↔
  ∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndepSets_singleton_iff
theorem condIndepSets_singleton_iff {μ : Measure Ω} [IsFiniteMeasure μ]
    {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t) :
    CondIndepSets m' hm' {s} {t} μ ↔ (μ⟦s ∩ t | m'⟧) =ᵐ[μ] (μ⟦s | m'⟧) * (μ⟦t | m'⟧) 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] {s t : Set Ω} (hs : MeasurableSet s) (ht : MeasurableSet t)
Goal: ProbabilityTheory.CondIndepSets m' hm' {s} {t} μ ↔
  μ[(s ∩ t).indicator fun ω => 1|m'] =ᵐ[μ] μ[s.indicator fun ω => 1|m'] * μ[t.indicator fun ω => 1|m']
-- ProbabilityTheory.iCondIndep_iff_iCondIndepSets
lemma iCondIndep_iff_iCondIndepSets (m : ι → MeasurableSpace Ω)
    (μ : @Measure Ω mΩ) [IsFiniteMeasure μ] :
    iCondIndep m' hm' m μ ↔ iCondIndepSets m' hm' (fun x ↦ {s | MeasurableSet[m x] s}) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (m : ι → MeasurableSpace Ω) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndep m' hm' m μ ↔ ProbabilityTheory.iCondIndepSets m' hm' (fun x => {s | MeasurableSet s}) μ
-- ProbabilityTheory.iCondIndep_iff
lemma iCondIndep_iff (m : ι → MeasurableSpace Ω) (hm : ∀ i, m i ≤ mΩ)
    (μ : @Measure Ω mΩ) [IsFiniteMeasure μ] :
    iCondIndep m' hm' m μ
      ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[m i] (f i)),
      μ⟦⋂ i ∈ s, f i | m'⟧ =ᵐ[μ] ∏ i ∈ s, (μ⟦f i | m'⟧) 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (m : ι → MeasurableSpace Ω) (hm : ∀ (i : ι), m i ≤ mΩ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndep m' hm' m μ ↔
  ∀ (s : Finset ι) {f : ι → Set Ω},
    (∀ i ∈ s, MeasurableSet (f i)) →
      μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndep_iff_condIndepSets
lemma condIndep_iff_condIndepSets (m' m₁ m₂ : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω}
    [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (μ : Measure Ω ) [IsFiniteMeasure μ] :
    CondIndep m' m₁ m₂ hm' μ
      ↔ CondIndepSets m' hm' {s | MeasurableSet[m₁] s} {s | MeasurableSet[m₂] s} μ 
Arguments: {Ω : Type u_1} (m' m₁ m₂ : MeasurableSpace Ω) {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 ↔
  ProbabilityTheory.CondIndepSets m' hm' {s | MeasurableSet s} {s | MeasurableSet s} μ
-- ProbabilityTheory.condIndep_iff
lemma condIndep_iff (m' m₁ m₂ : MeasurableSpace Ω)
    {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω]
    (hm' : m' ≤ mΩ) (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ) (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndep m' m₁ m₂ hm' μ
      ↔ ∀ t1 t2, MeasurableSet[m₁] t1 → MeasurableSet[m₂] t2
        → (μ⟦t1 ∩ t2 | m'⟧) =ᵐ[μ] (μ⟦t1 | m'⟧) * (μ⟦t2 | m'⟧) 
Arguments: {Ω : Type u_1} (m' m₁ m₂ : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndep m' m₁ m₂ hm' μ ↔
  ∀ (t1 t2 : Set Ω),
    MeasurableSet t1 →
      MeasurableSet t2 →
        μ[(t1 ∩ t2).indicator fun ω => 1|m'] =ᵐ[μ] μ[t1.indicator fun ω => 1|m'] * μ[t2.indicator fun ω => 1|m']
-- ProbabilityTheory.iCondIndepSet_iff_iCondIndep
lemma iCondIndepSet_iff_iCondIndep (s : ι → Set Ω) (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepSet m' hm' s μ ↔ iCondIndep m' hm' (fun i ↦ generateFrom {s i}) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s : ι → Set Ω) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepSet m' hm' s μ ↔
  ProbabilityTheory.iCondIndep m' hm' (fun i => MeasurableSpace.generateFrom {s i}) μ
-- ProbabilityTheory.iCondIndepSet_iff_iCondIndepSets_singleton
theorem iCondIndepSet_iff_iCondIndepSets_singleton (s : ι → Set Ω) (hs : ∀ i, MeasurableSet (s i))
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepSet m' hm' s μ ↔ iCondIndepSets m' hm' (fun i ↦ {s i}) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s : ι → Set Ω) (hs : ∀ (i : ι), MeasurableSet (s i)) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepSet m' hm' s μ ↔ ProbabilityTheory.iCondIndepSets m' hm' (fun i => {s i}) μ
-- ProbabilityTheory.iCondIndepSet_iff
lemma iCondIndepSet_iff (s : ι → Set Ω) (hs : ∀ i, MeasurableSet (s i))
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepSet m' hm' s μ ↔
      ∀ S : Finset ι, μ⟦⋂ i ∈ S, s i | m'⟧ =ᵐ[μ] ∏ i ∈ S, μ⟦s i | m'⟧ 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s : ι → Set Ω) (hs : ∀ (i : ι), MeasurableSet (s i)) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepSet m' hm' s μ ↔
  ∀ (S : Finset ι), μ[(⋂ i ∈ S, s i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ S, μ[(s i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndepSet_iff_condIndep
lemma condIndepSet_iff_condIndep (s t : Set Ω) (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepSet m' hm' s t μ ↔ CondIndep m' (generateFrom {s}) (generateFrom {t}) hm' μ 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s t : Set Ω) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepSet m' hm' s t μ ↔
  ProbabilityTheory.CondIndep m' (MeasurableSpace.generateFrom {s}) (MeasurableSpace.generateFrom {t}) hm' μ
-- ProbabilityTheory.condIndepSet_iff_condIndepSets_singleton
theorem condIndepSet_iff_condIndepSets_singleton {s t : Set Ω} (hs_meas : MeasurableSet s)
    (ht_meas : MeasurableSet t) (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepSet m' hm' s t μ ↔ CondIndepSets m' hm' {s} {t} μ 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {s t : Set Ω} (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepSet m' hm' s t μ ↔ ProbabilityTheory.CondIndepSets m' hm' {s} {t} μ
-- ProbabilityTheory.condIndepSet_iff
lemma condIndepSet_iff (s t : Set Ω) (hs : MeasurableSet s) (ht : MeasurableSet t)
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepSet m' hm' s t μ ↔ (μ⟦s ∩ t | m'⟧) =ᵐ[μ] (μ⟦s | m'⟧) * (μ⟦t | m'⟧) 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) (s t : Set Ω) (hs : MeasurableSet s) (ht : MeasurableSet t) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepSet m' hm' s t μ ↔
  μ[(s ∩ t).indicator fun ω => 1|m'] =ᵐ[μ] μ[s.indicator fun ω => 1|m'] * μ[t.indicator fun ω => 1|m']
-- ProbabilityTheory.iCondIndepFun_iff_iCondIndep
lemma iCondIndepFun_iff_iCondIndep {β : ι → Type*}
    (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x)
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepFun m' hm' m f μ
      ↔ iCondIndep m' hm' (fun x ↦ MeasurableSpace.comap (f x) (m x)) μ 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {β : ι → Type u_3} (m : (x : ι) → MeasurableSpace (β x)) (f : (x : ι) → Ω → β x) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepFun m' hm' m f μ ↔
  ProbabilityTheory.iCondIndep m' hm' (fun x => MeasurableSpace.comap (f x) (m x)) μ
-- ProbabilityTheory.iCondIndepFun_iff
lemma iCondIndepFun_iff {β : ι → Type*}
    (m : ∀ x : ι, MeasurableSpace (β x)) (f : ∀ x : ι, Ω → β x) (hf : ∀ i, Measurable (f i))
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    iCondIndepFun m' hm' m f μ
      ↔ ∀ (s : Finset ι) {g : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[(m i).comap (f i)] (g i)),
      μ⟦⋂ i ∈ s, g i | m'⟧ =ᵐ[μ] ∏ i ∈ s, (μ⟦g i | m'⟧) 
Arguments: {Ω : Type u_1} {ι : Type u_2} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {β : ι → Type u_3} (m : (x : ι) → MeasurableSpace (β x)) (f : (x : ι) → Ω → β x) (hf : ∀ (i : ι), Measurable (f i)) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.iCondIndepFun m' hm' m f μ ↔
  ∀ (s : Finset ι) {g : ι → Set Ω},
    (∀ i ∈ s, MeasurableSet (g i)) →
      μ[(⋂ i ∈ s, g i).indicator fun ω => 1|m'] =ᵐ[μ] ∏ i ∈ s, μ[(g i).indicator fun ω => 1|m']
-- ProbabilityTheory.condIndepFun_iff_condIndep
lemma condIndepFun_iff_condIndep {β γ : Type*} [mβ : MeasurableSpace β]
    [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepFun m' hm' f g μ
      ↔ CondIndep m' (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) hm' μ 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {β : Type u_3} {γ : Type u_4} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
  ProbabilityTheory.CondIndep m' (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) hm' μ
-- ProbabilityTheory.condIndepFun_iff
lemma condIndepFun_iff {β γ : Type*} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ]
    (f : Ω → β) (g : Ω → γ) (hf : Measurable f) (hg : Measurable g)
    (μ : Measure Ω) [IsFiniteMeasure μ] :
    CondIndepFun m' hm' f g μ ↔ ∀ t1 t2, MeasurableSet[MeasurableSpace.comap f mβ] t1
      → MeasurableSet[MeasurableSpace.comap g mγ] t2
        → (μ⟦t1 ∩ t2 | m'⟧) =ᵐ[μ] (μ⟦t1 | m'⟧) * (μ⟦t2 | m'⟧) 
Arguments: {Ω : Type u_1} (m' : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω} [StandardBorelSpace Ω] [Nonempty Ω] (hm' : m' ≤ mΩ) {β : Type u_3} {γ : Type u_4} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (hf : Measurable f) (hg : Measurable g) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ]
Goal: ProbabilityTheory.CondIndepFun m' hm' f g μ ↔
  ∀ (t1 t2 : Set Ω),
    MeasurableSet t1 →
      MeasurableSet t2 →
        μ[(t1 ∩ t2).indicator fun ω => 1|m'] =ᵐ[μ] μ[t1.indicator fun ω => 1|m'] * μ[t2.indicator fun ω => 1|m']
-- ProbabilityTheory.CondIndepSets.symm
@[symm]
theorem CondIndepSets.symm {s₁ s₂ : Set (Set Ω)}
    (h : CondIndepSets m' hm' s₁ s₂ μ) : CondIndepSets m' hm' s₂ s₁ 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 : ProbabilityTheory.CondIndepSets m' hm' s₁ s₂ μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' s₂ s₁ μ
-- ProbabilityTheory.condIndepSets_of_condIndepSets_of_le_left
theorem condIndepSets_of_condIndepSets_of_le_left {s₁ s₂ s₃ : Set (Set Ω)}
    (h_indep : CondIndepSets m' hm' s₁ s₂ μ) (h31 : s₃ ⊆ s₁) :
    CondIndepSets m' hm' s₃ 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 s₃ : Set (Set Ω)} (h_indep : ProbabilityTheory.CondIndepSets m' hm' s₁ s₂ μ) (h31 : s₃ ⊆ s₁)
Goal: ProbabilityTheory.CondIndepSets m' hm' s₃ s₂ μ
-- ProbabilityTheory.condIndepSets_of_condIndepSets_of_le_right
theorem condIndepSets_of_condIndepSets_of_le_right {s₁ s₂ s₃ : Set (Set Ω)}
    (h_indep : CondIndepSets m' hm' s₁ s₂ μ) (h32 : s₃ ⊆ s₂) :
    CondIndepSets m' hm' s₁ s₃ 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 s₃ : Set (Set Ω)} (h_indep : ProbabilityTheory.CondIndepSets m' hm' s₁ s₂ μ) (h32 : s₃ ⊆ s₂)
Goal: ProbabilityTheory.CondIndepSets m' hm' s₁ s₃ μ
-- ProbabilityTheory.CondIndepSets.union
theorem CondIndepSets.union {s₁ s₂ s' : Set (Set Ω)}
    (h₁ : CondIndepSets m' hm' s₁ s' μ) (h₂ : CondIndepSets m' hm' s₂ 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 s₂ s' : Set (Set Ω)} (h₁ : ProbabilityTheory.CondIndepSets m' hm' s₁ s' μ) (h₂ : ProbabilityTheory.CondIndepSets m' hm' s₂ s' μ)
Goal: ProbabilityTheory.CondIndepSets m' hm' (s₁ ∪ s₂) s' μ
-- ProbabilityTheory.CondIndepSets.union_iff
@[simp]
theorem CondIndepSets.union_iff {s₁ s₂ s' : Set (Set Ω)}  :
    CondIndepSets m' hm' (s₁ ∪ s₂) s' μ
      ↔ CondIndepSets m' hm' s₁ s' μ ∧ CondIndepSets m' hm' 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 s₂ s' : Set (Set Ω)}
Goal: ProbabilityTheory.CondIndepSets m' hm' (s₁ ∪ s₂) s' μ ↔
  ProbabilityTheory.CondIndepSets m' hm' s₁ s' μ ∧ ProbabilityTheory.CondIndepSets m' hm' s₂ s' μ