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|
; -------------------------------------------------------------------------
; Boogie 2 universal background predicate for Z3 (Simplify notation with types)
; Copyright (c) 2004-2009, Microsoft Corp.
(DEFTYPE $int :BUILTIN Int)
(DEFTYPE $bool :BUILTIN bool)
(DEFTYPE U)
(DEFTYPE T)
(DEFOP <: U U $bool) ; used for translation with type premisses
(DEFOP <:: T U U $bool) ; used for translation with type arguments
(BG_PUSH (AND
; false is not true
(DISTINCT |@false| |@true|)
; we assume type correctness of the operations here
; a-l>=0 ==> (v ++ w:l)[a:b] = v[a-l:b-l]
(FORALL (v lv w lw lvw a b)
(QID bv:e:c1)
(PATS ($bv_extract ($bv_concat v lv w lw) lvw a b))
(IMPLIES
(>= (- a lw) 0)
(EQ ($bv_extract ($bv_concat v lv w lw) lvw a b) ($bv_extract v lv (- a lw) (- b lw)))))
; b<=l ==> (v ++ w:l)[a:b] = w[a:b]
(FORALL (v lv w lw lvw a b)
(QID bv:e:c2)
(PATS ($bv_extract ($bv_concat v lv w lw) lvw a b))
(IMPLIES
(<= b lw)
(EQ ($bv_extract ($bv_concat v lv w lw) lvw a b) ($bv_extract w lw a b))))
; v:l
; a>=x || b<=y ==> (v[x:l] ++ w ++ v[0:y])[a:b] = v[a:b]
(FORALL (v lv x lxv w lw lwy y a b)
(QID bv:e:c3)
(PATS
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
($bv_concat
w lw
($bv_extract v lv 0 y) y) lwy) lv a b))
(IMPLIES
(AND
(EQ lw (- x y))
(EQ lxv (- lv x))
(EQ lwy (+ w y))
(OR (>= a x) (<= b y)))
(EQ
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
($bv_concat
w lw
($bv_extract v lv 0 y) y) lwy) lv a b)
($bv_extract v lv a b))))
; a>=x ==> (v[x:l] ++ w)[a:b] = v[a:b]
(FORALL (v lv x lxv w a b)
(QID bv:e:c4)
(PATS
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
w x)
lv a b))
(IMPLIES
(AND
(EQ lxv (- lv x))
(>= a x))
(EQ
($bv_extract
($bv_concat
($bv_extract v lv x lv) lxv
w x)
lv a b)
($bv_extract v lv a b))))
(FORALL (v l)
(QID bv:e:0)
(PATS ($bv_extract v l 0 l))
(EQ ($bv_extract v l 0 l) v))
(FORALL (n)
(QID bv:pow)
(PATS ($pow2 n))
(IMPLIES (> n 0) (EQ ($pow2 n) (* 2 ($pow2 (- n 1))))))
(EQ ($pow2 0) 1)
; 0 <= v < 2^Y ==> 0bvX ++ v[0:Y] == v
(FORALL (v l a b)
(QID bv:e:pow)
(PATS ($bv_concat 0 b ($bv_extract v l 0 a) a))
(IMPLIES
(AND
(<= 0 v)
(< v ($pow2 a))
(EQ l (+ a b)))
(EQ ($bv_concat 0 b ($bv_extract v l 0 a) a) v)))
; X > 0 ==> 0bvX ++ v >= 0
(FORALL (v a b)
(QID bv:e:pos)
(PATS ($bv_concat 0 b v a))
(IMPLIES
(> b 0)
(>= ($bv_concat 0 b v a) 0)))
;; unsound?
; (FORALL (lv w lw)
; (QID bv:c:0)
; (PATS ($bv_concat 0 lv w lw))
; (EQ ($bv_concat 0 lv w lw) w))
;; matching loop
; (FORALL (v l1 a b l2 c d)
; (QID bv:e:e)
; (PATS ($bv_extract ($bv_extract v l1 a b) l2 c d))
; (EQ ($bv_extract ($bv_extract v l1 a b) l2 c d) ($bv_extract v l1 (+ c a) (+ d a))))
; Reflect plus
(FORALL (a b) (PATS (Reflect$Add a b)) (EQ (Reflect$Add a b) (+ a b)))
)) ;; AND, BG_PUSH
; End Boogie universal background predicate
; -------------------------------------------------------------------------
|
