Theory "bft"

Signature

Definitions

Rel_def

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

Theorems

BFT_def

⊢ FINITE (Parents G) ⇒
  BFT G f seen [] acc = acc ∧
  BFT G f seen (h::t) acc = if MEM h seen then BFT G f seen t acc
  else BFT G f (h::seen) (t ++ G h) (f h acc)

BFT_ind

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

BFT_CONS

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

BFT_FOLD

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

BFT_ALL_DISTINCT

⊢ ∀G seen fringe.
      FINITE (Parents G) ⇒ ALL_DISTINCT (BFT G CONS seen fringe [])

BFT_REACH_1

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

BFT_REACH_2

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

BFT_REACH_THM

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