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-// VSComp 2010, problem 3, find a 0 in a linked list and return how many nodes were skipped
-// until the first 0 (or end-of-list) was found.
-// Rustan Leino, 18 August 2010.
-//
-// The difficulty in this problem lies in specifying what the return value 'r' denotes and in
-// proving that the program terminates. Both of these are addressed by declaring a ghost
-// field 'List' in each linked-list node, abstractly representing the linked-list elements
-// from the node to the end of the linked list. The specification can now talk about that
-// sequence of elements and can use 'r' as an index into the sequence, and termination can
-// be proved from the fact that all sequences in Dafny are finite.
-//
-// We only want to deal with linked lists whose 'List' field is properly filled in (which
-// can only happen in an acyclic list, for example). To that avail, the standard idiom in
-// Dafny is to declare a predicate 'Valid()' that is true of an object when the data structure
-// representing object's abstract value is properly formed. The definition of 'Valid()'
-// is what one intuitively would think of as the ''object invariant'', and it is mentioned
-// explicitly in method pre- and postconditions. As part of this standard idiom, one also
-// declared a ghost variable 'Repr' that is maintained as the set of objects that make up
-// the representation of the aggregate object--in this case, the Node itself and all its
-// successors.
-
-class Node {
- ghost var List: seq<int>;
- ghost var Repr: set<Node>;
- var head: int;
- var next: Node;
-
- function Valid(): bool
- reads this, Repr;
- {
- this in Repr &&
- 1 <= |List| && List[0] == head &&
- (next == null ==> |List| == 1) &&
- (next != null ==>
- next in Repr && next.Repr <= Repr && this !in next.Repr && next.Valid() && next.List == List[1..])
- }
-
- static method Cons(x: int, tail: Node) returns (n: Node)
- requires tail == null || tail.Valid();
- ensures n != null && n.Valid();
- ensures if tail == null then n.List == [x] else n.List == [x] + tail.List;
- {
- n := new Node;
- n.head := x;
- n.next := tail;
- if (tail == null) {
- n.List := [x];
- n.Repr := {n};
- } else {
- n.List := [x] + tail.List;
- n.Repr := {n} + tail.Repr;
- }
- }
-}
-
-static method Search(ll: Node) returns (r: int)
- requires ll == null || ll.Valid();
- ensures ll == null ==> r == 0;
- ensures ll != null ==>
- 0 <= r && r <= |ll.List| &&
- (r < |ll.List| ==> ll.List[r] == 0 && 0 !in ll.List[..r]) &&
- (r == |ll.List| ==> 0 !in ll.List);
-{
- if (ll == null) {
- r := 0;
- } else {
- var jj := ll;
- var i := 0;
- while (jj != null && jj.head != 0)
- invariant jj != null ==> jj.Valid() && i + |jj.List| == |ll.List| && ll.List[i..] == jj.List;
- invariant jj == null ==> i == |ll.List|;
- invariant 0 !in ll.List[..i];
- decreases |ll.List| - i;
- {
- jj := jj.next;
- i := i + 1;
- }
- r := i;
- }
-}
-
-method Main()
-{
- var list: Node := null;
- list := list.Cons(0, list);
- list := list.Cons(5, list);
- list := list.Cons(0, list);
- list := list.Cons(8, list);
- var r := Search(list);
- print "Search returns ", r, "\n";
- assert r == 1;
-}