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class IntSet {
ghost var Contents: set<int>;
ghost var Repr: set<object>;
var root: Node;
function Valid(): bool
reads this, Repr;
{
this in Repr &&
(root == null ==> Contents == {}) &&
(root != null ==>
root in Repr && root.Repr <= Repr && this !in root.Repr &&
root.Valid() &&
Contents == root.Contents)
}
method Init()
modifies this;
ensures Valid() && fresh(Repr - {this});
ensures Contents == {};
{
root := null;
Repr := {this};
Contents := {};
}
method Find(x: int) returns (present: bool)
requires Valid();
ensures present <==> x in Contents;
{
if (root == null) {
present := false;
} else {
present := root.Find(x);
}
}
method Insert(x: int)
requires Valid();
modifies Repr;
ensures Valid() && fresh(Repr - old(Repr));
ensures Contents == old(Contents) + {x};
{
var t := InsertHelper(x, root);
root := t;
Contents := root.Contents;
Repr := root.Repr + {this};
}
static method InsertHelper(x: int, n: Node) returns (m: Node)
requires n == null || n.Valid();
modifies n.Repr;
ensures m != null && m.Valid();
ensures n == null ==> fresh(m.Repr) && m.Contents == {x};
ensures n != null ==> m == n && n.Contents == old(n.Contents) + {x};
ensures n != null ==> fresh(n.Repr - old(n.Repr));
decreases if n == null then {} else n.Repr;
{
if (n == null) {
m := new Node.Init(x);
} else if (x == n.data) {
m := n;
} else {
if (x < n.data) {
assert n.right == null || n.right.Valid();
var t := InsertHelper(x, n.left);
n.left := t;
n.Repr := n.Repr + n.left.Repr;
} else {
assert n.left == null || n.left.Valid();
var t := InsertHelper(x, n.right);
n.right := t;
n.Repr := n.Repr + n.right.Repr;
}
n.Contents := n.Contents + {x};
m := n;
}
}
method Remove(x: int)
requires Valid();
modifies Repr;
ensures Valid() && fresh(Repr - old(Repr));
ensures Contents == old(Contents) - {x};
{
if (root != null) {
var newRoot := root.Remove(x);
root := newRoot;
if (root == null) {
Contents := {};
Repr := {this};
} else {
Contents := root.Contents;
Repr := root.Repr + {this};
}
}
}
}
class Node {
ghost var Contents: set<int>;
ghost var Repr: set<object>;
var data: int;
var left: Node;
var right: Node;
function Valid(): bool
reads this, Repr;
{
this in Repr &&
null !in Repr &&
(left != null ==>
left in Repr &&
left.Repr <= Repr && this !in left.Repr &&
left.Valid() &&
(forall y :: y in left.Contents ==> y < data)) &&
(right != null ==>
right in Repr &&
right.Repr <= Repr && this !in right.Repr &&
right.Valid() &&
(forall y :: y in right.Contents ==> data < y)) &&
(left == null && right == null ==>
Contents == {data}) &&
(left != null && right == null ==>
Contents == left.Contents + {data}) &&
(left == null && right != null ==>
Contents == {data} + right.Contents) &&
(left != null && right != null ==>
left.Repr !! right.Repr &&
Contents == left.Contents + {data} + right.Contents)
}
method Init(x: int)
modifies this;
ensures Valid() && fresh(Repr - {this});
ensures Contents == {x};
{
data := x;
left := null;
right := null;
Contents := {x};
Repr := {this};
}
method Find(x: int) returns (present: bool)
requires Valid();
ensures present <==> x in Contents;
decreases Repr;
{
if (x == data) {
present := true;
} else if (left != null && x < data) {
present := left.Find(x);
} else if (right != null && data < x) {
present := right.Find(x);
} else {
present := false;
}
}
method Remove(x: int) returns (node: Node)
requires Valid();
modifies Repr;
ensures fresh(Repr - old(Repr));
ensures node != null ==> node.Valid();
ensures node == null ==> old(Contents) <= {x};
ensures node != null ==> node.Repr <= Repr && node.Contents == old(Contents) - {x};
decreases Repr;
{
node := this;
if (left != null && x < data) {
var t := left.Remove(x);
left := t;
Contents := Contents - {x};
if (left != null) { Repr := Repr + left.Repr; }
} else if (right != null && data < x) {
var t := right.Remove(x);
right := t;
Contents := Contents - {x};
if (right != null) { Repr := Repr + right.Repr; }
} else if (x == data) {
if (left == null && right == null) {
node := null;
} else if (left == null) {
node := right;
} else if (right == null) {
node := left;
} else {
// rotate
var min, r := right.RemoveMin();
data := min; right := r;
Contents := Contents - {x};
if (right != null) { Repr := Repr + right.Repr; }
}
}
}
method RemoveMin() returns (min: int, node: Node)
requires Valid();
modifies Repr;
ensures fresh(Repr - old(Repr));
ensures node != null ==> node.Valid();
ensures node == null ==> old(Contents) == {min};
ensures node != null ==> node.Repr <= Repr && node.Contents == old(Contents) - {min};
ensures min in old(Contents) && (forall x :: x in old(Contents) ==> min <= x);
decreases Repr;
{
if (left == null) {
min := data;
node := right;
} else {
var t;
min, t := left.RemoveMin();
left := t;
node := this;
Contents := Contents - {min};
if (left != null) { Repr := Repr + left.Repr; }
}
}
}
class Main {
method Client0(x: int)
{
var s := new IntSet.Init();
s.Insert(12);
s.Insert(24);
var present := s.Find(x);
assert present <==> x == 12 || x == 24;
}
method Client1(s: IntSet, x: int)
requires s != null && s.Valid();
modifies s.Repr;
{
s.Insert(x);
s.Insert(24);
assert old(s.Contents) - {x,24} == s.Contents - {x,24};
}
}
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