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class Queue<T> {
var head: Node<T>;
var tail: Node<T>;
var contents: seq<T>;
var footprint: set<object>;
var spine: set<Node<T>>;
function Valid(): bool
reads this, footprint;
{
this in footprint && spine <= footprint &&
head != null && head in spine &&
tail != null && tail in spine &&
tail.next == null &&
(forall n ::
n in spine ==>
n != null && n.Valid() &&
n.footprint <= footprint &&
(n.next == null ==> n == tail)) &&
(forall n ::
n in spine ==>
n.next != null ==> n.next in spine) &&
contents == head.tailContents
}
method Init()
modifies this;
ensures Valid() && fresh(footprint - {this});
ensures |contents| == 0;
{
var n := new Node<T>;
call n.Init();
head := n;
tail := n;
contents := n.tailContents;
footprint := {this} + n.footprint;
spine := {n};
}
method IsEmpty() returns (isEmpty: bool)
requires Valid();
ensures isEmpty <==> |contents| == 0;
{
isEmpty := head == tail;
}
method Enqueue(t: T)
requires Valid();
modifies footprint;
ensures Valid() && fresh(footprint - old(footprint));
ensures contents == old(contents) + [t];
{
var n := new Node<T>;
call n.Init(); n.data := t;
tail.next := n;
tail := n;
foreach (m in spine) {
m.tailContents := m.tailContents + [t];
}
contents := head.tailContents;
foreach (m in spine) {
m.footprint := m.footprint + n.footprint;
}
footprint := footprint + n.footprint;
spine := spine + {n};
}
method Front() returns (t: T)
requires Valid();
requires 0 < |contents|;
ensures t == contents[0];
{
t := head.next.data;
}
method Dequeue()
requires Valid();
requires 0 < |contents|;
modifies footprint;
ensures Valid() && fresh(footprint - old(footprint));
ensures contents == old(contents)[1..];
{
var n := head.next;
head := n;
contents := n.tailContents;
}
method Rotate()
requires Valid();
requires 0 < |contents|;
modifies footprint;
ensures Valid() && fresh(footprint - old(footprint));
ensures contents == old(contents)[1..] + old(contents)[..1];
{
var t: T;
call t := Front();
call Dequeue();
call Enqueue(t);
}
method RotateAny()
requires Valid();
requires 0 < |contents|;
modifies footprint;
ensures Valid() && fresh(footprint - old(footprint));
ensures (exists i :: 0 <= i && i <= |contents| &&
contents == old(contents)[i..] + old(contents)[..i]);
{
var t: T;
call t := Front();
call Dequeue();
call Enqueue(t);
}
}
class Node<T> {
var data: T;
var next: Node<T>;
var tailContents: seq<T>;
var footprint: set<object>;
function Valid(): bool
reads this, footprint;
{
this in footprint &&
(next != null ==> next in footprint && next.footprint <= footprint) &&
(next == null ==> tailContents == []) &&
(next != null ==> tailContents == [next.data] + next.tailContents)
}
method Init()
modifies this;
ensures Valid() && fresh(footprint - {this});
ensures next == null;
{
next := null;
tailContents := [];
footprint := {this};
}
}
class Main<U> {
method A<T>(t: T, u: T, v: T)
{
var q0 := new Queue<T>;
call q0.Init();
var q1 := new Queue<T>;
call q1.Init();
call q0.Enqueue(t);
call q0.Enqueue(u);
call q1.Enqueue(v);
assert |q0.contents| == 2;
var w;
call w := q0.Front();
assert w == t;
call q0.Dequeue();
call w := q0.Front();
assert w == u;
assert |q0.contents| == 1;
assert |q1.contents| == 1;
}
method Main2(t: U, u: U, v: U, q0: Queue<U>, q1: Queue<U>)
requires q0 != null && q0.Valid();
requires q1 != null && q1.Valid();
requires q0.footprint !! q1.footprint;
requires |q0.contents| == 0;
modifies q0.footprint, q1.footprint;
ensures fresh(q0.footprint - old(q0.footprint));
ensures fresh(q1.footprint - old(q1.footprint));
{
call q0.Enqueue(t);
call q0.Enqueue(u);
call q1.Enqueue(v);
assert |q0.contents| == 2;
var w;
call w := q0.Front();
assert w == t;
call q0.Dequeue();
call w := q0.Front();
assert w == u;
assert |q0.contents| == 1;
assert |q1.contents| == old(|q1.contents|) + 1;
}
}
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