Presburger Arithmetic and Its Use in Verification

B.5.
COOPER.FS (EXCERPT)
| SAnd(f, vr) −> (f |> cooperParallel, vr) |>SAnd
| SOr(f, vr) −> (f |> cooperParallel, vr) |>SOr
| E(xs, SOr(f, vr))
−> SOr(cooperParallel (E(xs, f)), vr)
| E(xs, Or fs) −> //printfn "EOr.xs=%i, fs=%i" xs.Length fs.Length
fs |> parMap (fun f −> cooperParallel (E(xs, f))) |> Or
| E(xs, f) −> List.fold (fun acc x −> (elimQuantifierParallel x acc)) (nnf (
cooperParallel f)) xs
| A(xs, SAnd(f, vr))
−> SAnd(cooperParallel (A(xs, f)), vr)
| A(xs, And fs) −> //printfn "AAnd.xs=%i, fs=%i" xs.Length fs.Length
fs |> parMap (fun f −> cooperParallel (A(xs, f))) |> And
| A(xs, f) −> Not (List.fold (fun acc x −> (elimQuantifierParallel x acc)) (
nnf (cooperParallel (Not f))) xs)
| _ −> formula
let elimQuantifiers = cooper >> reduce
let elimQuantifiersParallel = cooperParallel >> reduce
//
// Evaluation part
//
let cartesian lss =
let klls=[for x in l do
for xs in ls −> x::xs ]
List.foldBack k lss [[]]
let genRangeArray vr =
vr |> List.map (fun (v, r) −> List.init r (fun i −> (v, i))) |> cartesian |>
List.toArray
// Substitute a list of variables and associated values to a formula
// Suppose the formula is quantifier−free
let rec evalFormula xts formula =
match formula with
| C(t, ct) −> match subst (t, xts) with
| t’ −> if isConstTerm t’ then
let c = getConst t’
match ct, c with
| EQ, 0−> TT
| UEQ, xwhenx 0 −> TT
| GT, xwhenx>0−> TT
| _, _ −> FF
else invalidArg "evalFormula" (string xts)
| D(i, t) −> match subst (t, xts) with
| t’ −> if isConstTerm t’ then
let c = getConst t’
if i %| c then TT else FF
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