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// This file contains three variations of the separation-logic lseg linked-list example.
// In this first variation, the auxiliary information about the contents represented by a linked list
// and the subpart of the heap that the list occupies is passed around as additional parameters. In
// separation logic, the contents (here called 'q') would indeed be passed around in that way, whereas
// separating conjunction and the abstract predicate ListSegment would take care of staking out which
// subpart of the heap is being occupied by the linked-list representation.
class Node<T> {
var data: T;
var next: Node<T>;
static function ListSegment(q: seq<T>, from: Node<T>, to: Node<T>, S: set<Node<T>>): bool
reads S;
{
if q == []
then from == to
else from != null && from in S && from.data == q[0] && ListSegment(q[1..], from.next, to, S - {from})
}
static method Create(x: T) returns (l: Node<T>, ghost S: set<Node<T>>)
ensures ListSegment([x], l, null, S) && fresh(S);
{
// By using the following non-deterministic 'if' statement, the example shows that 'Create' can be
// implemented either directly or by call 'Cons'.
if (*) {
l := new Node<T>;
l.data := x;
l.next := null;
S := {l};
} else {
call l, S := Cons(x, null, [], {});
}
}
static method Cons(x: T, tail: Node<T>, ghost q: seq<T>, ghost S: set<Node<T>>) returns (l: Node<T>, ghost U: set<Node<T>>)
requires ListSegment(q, tail, null, S);
ensures ListSegment([x] + q, l, null, U) && fresh(U - S);
{
l := new Node<T>;
l.data := x;
l.next := tail;
U := S + {l};
}
}
// The following class is a variation of the one above. The difference is that, in this one, each node
// keeps track of its own contents (called 'q' above) and representation (called 'S' and 'U' above).
class ListNode<T> {
ghost var Contents: seq<T>;
ghost var Repr: set<ListNode<T>>;
var data: T;
var next: ListNode<T>;
static function IsList(l: ListNode<T>): bool
reads l, l.Repr;
{
if l == null then
true
else if l.next == null then
l in l.Repr && l.Contents == [l.data]
else
{l, l.next} <= l.Repr && l.Contents == [l.data] + l.next.Contents && l.next.Repr <= l.Repr - {l} && IsList(l.next)
}
static method Create(x: T) returns (l: ListNode<T>)
ensures IsList(l) && l != null && l.Contents == [x] && fresh({l} + l.Repr);
{
// By using the following non-deterministic 'if' statement, the example shows that 'Create' can be
// implemented either directly or by call 'Cons'.
if (*) {
l := new ListNode<T>;
l.data := x;
l.next := null;
l.Repr := {l};
l.Contents := [x];
} else {
call l := Cons(x, null);
}
}
static method Cons(x: T, tail: ListNode<T>) returns (l: ListNode<T>)
requires IsList(tail);
ensures IsList(l) && l != null;
ensures tail == null ==> l.Contents == [x] && fresh({l} + l.Repr);
ensures tail != null ==> l.Contents == [x] + tail.Contents && fresh({l} + l.Repr - tail.Repr);
{
l := new ListNode<T>;
l.data := x;
l.next := tail;
if (tail != null) {
l.Repr := tail.Repr + {l};
l.Contents := [x] + tail.Contents;
} else {
l.Repr := {l};
l.Contents := [x];
}
}
}
// In this final variation, a list, empty or not, is represented by a List object. The representation
// of a List object includes a number of linked-list nodes. To make the treatment of the empty list
// nicer than in the class above, the List object starts its list with a sentinel object whose 'data'
// field is not used.
class List<T>
{
ghost var Contents: seq<T>;
ghost var Repr: set<object>;
var head: LLNode<T>;
function IsList(): bool
reads this, Repr;
{
this in Repr && head != null && head in Repr &&
head.Repr <= Repr && this !in head.Repr && head.IsWellFormed() &&
Contents == head.TailContents
}
method Init()
modifies this;
ensures IsList() && Contents == [] && fresh(Repr - {this});
{
var h := new LLNode<T>;
h.next := null;
h.TailContents := [];
h.Repr := {h};
head := h;
Contents := [];
Repr := {this} + h.Repr;
}
method Cons(x: T)
requires IsList();
modifies Repr;
ensures IsList() && Contents == [x] + old(Contents) && fresh(Repr - old(Repr));
{
head.data := x;
assert head.IsWellFormed(); // head remains well-formed even after assigning to an object (namely, head) in head.Repr
var h := new LLNode<T>;
h.next := head;
h.TailContents := [x] + head.TailContents;
h.Repr := {h} + head.Repr;
head := h;
Contents := [x] + Contents;
Repr := Repr + {h};
}
}
class LLNode<T>
{
var data: T;
var next: LLNode<T>;
ghost var TailContents: seq<T>;
ghost var Repr: set<object>;
function IsWellFormed(): bool
reads this, Repr;
{
this in Repr &&
(next == null ==> TailContents == []) &&
(next != null ==>
next in Repr &&
next.Repr <= Repr && this !in next.Repr && next.IsWellFormed() &&
TailContents == [next.data] + next.TailContents)
}
}
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