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|
datatype List<T> = Nil | Cons(T, List<T>);
class Node {
var data: int;
var next: Node;
function Repr(list: List<int>): bool
reads *;
decreases list;
{ match list
case Nil =>
next == null
case Cons(d,cdr) =>
data == d && next != null && next.Repr(cdr)
}
method Init()
modifies this;
ensures Repr(Nil);
{
next := null;
}
method Add(d: int, L: List<int>) returns (r: Node)
requires Repr(L);
ensures r != null && r.Repr(Cons(d, L));
{
r := new Node;
r.data := d;
r.next := this;
}
}
class AnotherNode {
var data: int;
var next: AnotherNode;
function Repr(n: AnotherNode, list: List<int>): bool
reads *;
decreases list;
{ match list
case Nil =>
n == null
case Cons(d,cdr) =>
n != null && n.data == d && Repr(n.next, cdr)
}
method Create() returns (n: AnotherNode)
ensures Repr(n, Nil);
{
n := null;
}
method Add(n: AnotherNode, d: int, L: List<int>) returns (r: AnotherNode)
requires Repr(n, L);
ensures Repr(r, Cons(d, L));
{
r := new AnotherNode;
r.data := d;
r.next := n;
}
}
method TestAllocatednessAxioms(a: List<Node>, b: List<Node>, c: List<AnotherNode>)
{
var n := new Node;
var p := n;
match a {
case Nil =>
case Cons(x, tail) => assert x != n; p := x;
}
match b {
case Nil =>
case Cons(x, tail) =>
match tail {
case Nil =>
case Cons(y, more) =>
assert y != n;
assert y != p; // error: if p is car(a), then it and y may very well be equal
}
}
match c {
case Nil =>
case Cons(x, tail) =>
match tail {
case Nil =>
case Cons(y, more) =>
var o: object := y;
assert p != null ==> p != o; // follows from well-typedness
}
}
}
class NestedMatchExpr {
function Cadr<T>(a: List<T>, default: T): T
{
match a
case Nil => default
case Cons(x,t) =>
match t
case Nil => default
case Cons(y,tail) => y
}
// CadrAlt is the same as Cadr, but it writes its two outer cases in the opposite order
function CadrAlt<T>(a: List<T>, default: T): T
{
match a
case Cons(x,t) => (
match t
case Nil => default
case Cons(y,tail) => y)
case Nil => default
}
method TestNesting0()
{
var x := 5;
var list := Cons(3, Cons(6, Nil));
assert Cadr(list, x) == 6;
match (list) {
case Nil => assert false;
case Cons(h,t) => assert Cadr(t, x) == 5;
}
}
method TestNesting1(a: List<NestedMatchExpr>)
ensures Cadr(a, this) == CadrAlt(a, this);
{
match (a) {
case Nil =>
case Cons(x,t) =>
match (t) {
case Nil =>
case Cons(y,tail) =>
}
}
}
}
// ------------------- datatype destructors ---------------------------------------
datatype XList = XNil | XCons(Car: int, Cdr: XList);
method Destructors0(d: XList) {
Lemma_AllCases(d);
if {
case d.XNil? =>
assert d == XNil;
case d.XCons? =>
var hd := d.Car;
var tl := d.Cdr;
assert d == XCons(hd, tl);
}
}
method Destructors1(d: XList) {
match (d) {
case XNil =>
assert d.XNil?;
case XCons(hd,tl) =>
assert d.XCons?;
}
}
method Destructors2(d: XList) {
// this method gets it backwards
match (d) {
case XNil =>
assert d.XCons?; // error
case XCons(hd,tl) =>
assert d.XNil?; // error
}
}
ghost method Lemma_AllCases(d: XList)
ensures d.XNil? || d.XCons?;
{
match (d) {
case XNil =>
case XCons(hd,tl) =>
}
}
method InjectivityTests(d: XList)
requires d != XNil;
{
match (d) {
case XCons(a,b) =>
match (d) {
case XCons(x,y) =>
assert a == x && b == y;
}
assert a == d.Car;
assert b == d.Cdr;
assert d == XCons(d.Car, d.Cdr);
}
}
|
