Member of a bitvector in Z3 - z3

Question

I need to know how to do the following in z3:
forall x in L4 . x < l2
forall x in L0 and forall y in L1. x < y
(declare-const L4 (_ BitVec 6))
(declare-const L1 (_ BitVec 6))
(declare-const L0 (_ BitVec 6))
(declare-const l2 Int)
(declare-const l1 Int)
assert L0 and L1 such that x < y
(check-sat)
Result is Sat
assert L0 and L1 such that y < x
(check-sat)
Result is UnSat

Answer 1

I'm assuming you are using bit-vectors to encode finite sets.
So, here is a possible encoding (it is also available online here):
;; You use bit-vectors of size 6. That is, we are encoding sets that are subsets of {0, 1, 2, ... 5}
(declare-const L4 (_ BitVec 6))
(declare-const L1 (_ BitVec 6))
(declare-const L0 (_ BitVec 6))
;; I will also encode l1 and l2 as Bit-Vectors.
;; Thus, I don't need to convert them into Bit-vectors to be able to use them in Bit-vector operations.
(declare-const l1 (_ BitVec 6))
(declare-const l2 (_ BitVec 6))
;; To make the problems more interesting let us assume that l1 and l2 are <= 5
(assert (bvsle l1 (_ bv5 6)))
(assert (bvsle l2 (_ bv5 6)))
;; All is encoding the set that contains {0, 1, 2, 3, 4, 5}
(define-const All (_ BitVec 6) #b111111)
;; Empty is encoding the empty set
(define-const Empty (_ BitVec 6) #b000000)
(define-fun LT_l ((S (_ BitVec 6)) (l (_ BitVec 6))) Bool
;; True if for all x in S x < l
;; Note that: All << l is the set of values greater than l
;; In SMT-LIB, bvshl is the left shift operator <<
(= (bvand (bvshl All l) S) Empty))
;; We can use shifting to encode the set that contains all values in {0, 1, 2, 3, 4, 5} that greater than or equal to l2
(define-const GE_l2 (_ BitVec 6) (bvshl All l2))
;; To assert that forall x in L4. x < l2, we assert that the intersection of L4 and GE_l2 is empty
(assert (= (bvand GE_l2 L4) Empty))
(check-sat)
(get-model)
(define-fun is_in ((e (_ BitVec 6)) (S (_ BitVec 6))) Bool
;; True if e is an element of the "set" S.
;; That is, it checks whether the (1 << e) && S is 0
(not (= (bvand (bvshl (_ bv1 6) e) S) Empty)))
(echo "is_in tests")
;; Test: is 2 in the set {0,3}
(simplify (is_in (_ bv2 6) #b001001))
;; Test: is 0 in the set {0, 3}
(simplify (is_in (_ bv0 6) #b001001))
;; Test: is 3 in the set {0, 3}
(simplify (is_in (_ bv3 6) #b001001))
;; Test: is 4 in the set {0, 3}
(simplify (is_in (_ bv4 6) #b001001))
(echo "end is_in tests")
(define-fun is_minimal ((e (_ BitVec 6)) (S (_ BitVec 6))) Bool
;; True if e is the minimal element of S
(and (is_in e S) ;; S contains e
;; (1 << e) - 1 represents the set of elements that are smaller than e
(= (bvand (bvsub (bvshl (_ bv1 6) e) (_ bv1 6)) S) Empty)))
(echo "is_minimal tests")
;; is 2 the minimal element in {5, 3}
(simplify (is_minimal (_ bv2 6) #b101000))
;; is 3 the minimal element in {5, 3}
(simplify (is_minimal (_ bv3 6) #b101000))
;; is 4 the minimal element in {5, 3}
(simplify (is_minimal (_ bv4 6) #b101000))
;; is 0 the minimal element in {4, 0}
(simplify (is_minimal (_ bv0 6) #b010001))
;; is 4 the minimal element in {4, 0}
(simplify (is_minimal (_ bv4 6) #b010001))
(echo "end is_minimal tests")
;; To encode that forall x in L0 and forall y in L1. x < y
(define-fun LT ((L0 (_ BitVec 6)) (L1 (_ BitVec 6))) Bool
; True if forall x in L0 and forall y in L1, x < y
; We encode that by saying that
; - L0 is empty OR
; - L1 is empty OR
; - Or Let min be the minimal value of L1 and Forall x in L0, x < min
(or (= L0 Empty)
(= L1 Empty)
(exists ((min (_ BitVec 6))) (and (is_minimal min L1) (LT_l L0 min)))))
;; To make the problem non-trivial, let us assume that L0 and L1 are not empty
(assert (not (= L0 Empty)))
(assert (not (= L1 Empty)))
(assert (LT L0 L1))
(check-sat)
(get-model)
(assert (LT L1 L0))
(check-sat)