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|
class Termination {
method A(N: int)
requires 0 <= N;
{
var i := 0;
while (i < N)
invariant i <= N;
// this will be heuristically inferred: decreases N - i;
{
i := i + 1;
}
}
method B(N: int)
requires 0 <= N;
{
var i := N;
while (true)
invariant 0 <= i;
decreases i;
{
i := i - 1;
if (!(0 <= i)) {
break;
}
}
assert i == -1;
}
method Lex() {
call x := Update();
call y := Update();
while (!(x == 0 && y == 0))
invariant 0 <= x && 0 <= y;
decreases x, y;
{
if (0 < y) {
y := y - 1;
} else {
x := x - 1;
call y := Update();
}
}
}
method Update() returns (r: int)
ensures 0 <= r;
{
r := 8;
}
method M() {
var b := true;
var i := 500;
var r := new Termination;
var s := {12, 200};
var q := [5, 8, 13];
while (true)
decreases b, i, r;
// invariant b ==> 0 <= i;
decreases s, q;
{
if (12 in s) {
s := s - {12};
} else if (b) {
b := !b;
i := i + 1;
} else if (20 <= i) {
i := i - 20;
} else if (r != null) {
r := null;
} else if (|q| != 0) {
q := q[1..];
} else {
break;
}
}
}
method Q<T>(list: List<T>) {
var l := list;
while (l != #List.Nil)
decreases l;
{
call x, l := Traverse(l);
}
}
method Traverse<T>(a: List<T>) returns (val: T, b: List<T>);
requires a != #List.Nil;
ensures a == #List.Cons(val, b);
}
datatype List<T> {
Nil;
Cons(T, List<T>);
}
method FailureToProveTermination0(N: int)
{
var n := N;
while (n < 100) { // error: may not terminate
n := n - 1;
}
}
method FailureToProveTermination1(x: int, y: int, N: int)
{
var n := N;
while (x < y && n < 100) // error: cannot prove termination from the heuristically chosen termination metric
{
n := n + 1;
}
}
method FailureToProveTermination2(x: int, y: int, N: int)
{
var n := N;
while (x < y && n < 100) // error: cannot prove termination from the given (bad) termination metric
decreases n - x;
{
n := n + 1;
}
}
method FailureToProveTermination3(x: int, y: int, N: int)
{
var n := N;
while (x < y && n < 100)
decreases 100 - n;
{
n := n + 1;
}
}
method FailureToProveTermination4(x: int, y: int, N: int)
{
var n := N;
while (n < 100 && x < y)
decreases 100 - n;
{
n := n + 1;
}
}
method FailureToProveTermination5(b: bool, N: int)
{
var n := N;
while (b && n < 100) // here, the heuristics are good enough to prove termination
{
n := n + 1;
}
}
class Node {
var next: Node;
var footprint: set<Node>;
function Valid(): bool
reads this, footprint;
// In a previous (and weaker) axiomatization of sets, there had been two problems
// with trying to prove the termination of this function. First, the default
// decreases clause (derived from the reads clause) had been done incorrectly for
// a list of expressions. Second, the set axiomatization had not been strong enough.
{
this in footprint && !(null in footprint) &&
(next != null ==> next in footprint &&
next.footprint < footprint &&
this !in next.footprint &&
next.Valid())
}
}
method DecreasesYieldsAnInvariant(z: int) {
var x := 100;
var y := 1;
var z := z; // make parameter into a local variable
while (x != y)
// inferred: decreases |x - y|;
invariant (0 < x && 0 < y) || (x < 0 && y < 0);
{
if (z == 52) {
break;
} else if (x < y) {
y := y - 1;
} else {
x := x - 1;
}
x := -x;
y := -y;
z := z + 1;
}
assert x - y < 100; // follows from the fact that no loop iteration increases what's in the 'decreases' clause
}
|
