Theory "dft"

Signature

Definitions

Rel_def

⊢ ∀G f seen to_visit acc.
    Rel (G,f,seen,to_visit,acc) =
    (CARD (Parents G DIFF LIST_TO_SET seen),LENGTH to_visit)

Theorems

DFT_ALL_DISTINCT

⊢ ∀G seen to_visit.
    FINITE (Parents G) ⇒ ALL_DISTINCT (DFT G CONS seen to_visit [])

DFT_CONS

⊢ ∀G f seen to_visit acc a b.
    FINITE (Parents G) ∧ (f = CONS) ∧ (acc = a ++ b) ⇒
    (DFT G f seen to_visit acc = DFT G f seen to_visit a ++ b)

DFT_FOLD

⊢ ∀G f seen to_visit acc.
    FINITE (Parents G) ⇒
    (DFT G f seen to_visit acc = FOLDR f acc (DFT G CONS seen to_visit []))

DFT_REACH_1

⊢ ∀G f seen to_visit acc.
    FINITE (Parents G) ∧ (f = CONS) ⇒
    ∀x. MEM x (DFT G f seen to_visit acc) ⇒
        x ∈ REACH_LIST G to_visit ∨ MEM x acc

DFT_REACH_2

⊢ ∀G f seen to_visit acc x.
    FINITE (Parents G) ∧ (f = CONS) ∧
    x ∈ REACH_LIST (EXCLUDE G (LIST_TO_SET seen)) to_visit ∧ ¬MEM x seen ⇒
    MEM x (DFT G f seen to_visit acc)

DFT_REACH_THM

⊢ ∀G to_visit.
    FINITE (Parents G) ⇒
    ∀x. x ∈ REACH_LIST G to_visit ⇔ MEM x (DFT G CONS [] to_visit [])

DFT_def

⊢ FINITE (Parents G) ⇒
  (DFT G f seen [] acc = acc) ∧
  (DFT G f seen (visit_now::visit_later) acc =
   if MEM visit_now seen then DFT G f seen visit_later acc
   else
     DFT G f (visit_now::seen) (G visit_now ++ visit_later) (f visit_now acc))

DFT_ind

⊢ ∀P. (∀G f seen visit_now visit_later acc.
         P G f seen [] acc ∧
         ((FINITE (Parents G) ∧ ¬MEM visit_now seen ⇒
           P G f (visit_now::seen) (G visit_now ++ visit_later)
             (f visit_now acc)) ∧
          (FINITE (Parents G) ∧ MEM visit_now seen ⇒
           P G f seen visit_later acc) ⇒
          P G f seen (visit_now::visit_later) acc)) ⇒
      ∀v v1 v2 v3 v4. P v v1 v2 v3 v4