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// ----- Stream
codatatype Stream = Nil | Cons(head: int, tail: Stream);
function append(M: Stream, N: Stream): Stream
{
match M
case Nil => N
case Cons(t, M') => Cons(t, append(M', N))
}
function zeros(): Stream
{
Cons(0, zeros())
}
function ones(): Stream
{
Cons(1, ones())
}
copredicate atmost(a: Stream, b: Stream)
{
match a
case Nil => true
case Cons(h,t) => b.Cons? && h <= b.head && atmost(t, b.tail)
}
comethod Theorem0()
ensures atmost(zeros(), ones());
{
// the following shows two equivalent ways to getting essentially the
// co-inductive hypothesis
if (*) {
Theorem0#[_k-1]();
} else {
Theorem0();
}
}
ghost method Theorem0_Manual()
ensures atmost(zeros(), ones());
{
parallel (k: nat) {
Theorem0_Lemma(k);
}
assume (forall k: nat :: atmost#[k](zeros(), ones())) ==> atmost(zeros(), ones());
}
datatype Natural = Zero | Succ(Natural);
comethod Theorem0_TerminationFailure_ExplicitDecreases(y: Natural)
ensures atmost(zeros(), ones());
decreases y;
{
match (y) {
case Succ(x) =>
// this is just to show that the decreases clause does kick in
Theorem0_TerminationFailure_ExplicitDecreases#[_k](x);
case Zero =>
Theorem0_TerminationFailure_ExplicitDecreases#[_k](y); // error: failure to terminate
}
Theorem0_TerminationFailure_ExplicitDecreases#[_k-1](y);
}
comethod Theorem0_TerminationFailure_DefaultDecreases(y: Natural)
ensures atmost(zeros(), ones());
{
match (y) {
case Succ(yy) =>
// this is just to show that the decreases clause does kick in
Theorem0_TerminationFailure_DefaultDecreases#[_k](yy);
case Zero =>
Theorem0_TerminationFailure_DefaultDecreases#[_k](y); // error: failure to terminate
}
Theorem0_TerminationFailure_DefaultDecreases#[_k-1](y);
}
ghost method {:induction true} Theorem0_Lemma(k: nat)
ensures atmost#[k](zeros(), ones());
{
}
comethod Theorem1()
ensures append(zeros(), ones()) == zeros();
{
Theorem1();
}
codatatype IList = ICons(head: int, tail: IList);
function UpIList(n: int): IList
{
ICons(n, UpIList(n+1))
}
copredicate Pos(s: IList)
{
s.head > 0 && Pos(s.tail)
}
comethod Theorem2(n: int)
requires 1 <= n;
ensures Pos(UpIList(n));
{
Theorem2(n+1);
}
comethod Theorem2_NotAProof(n: int)
requires 1 <= n;
ensures Pos(UpIList(n));
{ // error: this is not a proof
}
codatatype TList<T> = TCons(head: T, tail: TList);
function Next<T>(t: T): T
function FF<T>(h: T): TList<T>
{
TCons(h, FF(Next(h)))
}
function GG<T>(h: T): TList<T>
{
TCons(h, GG(Next(h)))
}
comethod Compare<T>(h: T)
ensures FF(h) == GG(h);
{
assert FF(h).head == GG(h).head;
Compare(Next(h));
if {
case true =>
assert FF(h).tail == GG(h).tail; // error: full equality is not known here
case true =>
// assert FF(h).tail == GG(h).tail;
case true =>
// assert FF(h).tail ==#[_k] GG(h).tail;
case true =>
// assert FF(h).tail ==#[_k - 1] GG(h).tail;
case true =>
}
}
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