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+// Schorr-Waite algorithm in Chalice
+// (see Test/dafny1/SchorrWaite.dfy)
+
+// (incomplete version) Two children instead of a sequence of an array
+class Node {
+ var left: Node;
+ var right: Node;
+ var marked: bool;
+ var l: bool;
+ var r: bool;
+}
+
+class Main {
+ method RecursiveMark(root: Node, S: seq<Node>)
+ requires acc(S[*].marked, 50) && acc(S[*].*, 50);
+ requires root != null && root in S;
+ // S is closed under 'left' and 'right':
+ requires forall n in S :: n != null &&
+ ((n.left != null ==> n.left in S) &&
+ (n.right != null ==> n.right in S));
+ requires forall n in S :: ! n.marked;
+ ensures acc(S[*].marked, 50) && acc(S[*].*, 50);
+ ensures root.marked;
+ // nodes reachable from 'root' are marked:
+ ensures forall n in S :: n.marked ==>
+ ((n.left != null ==> n.left in S && n.left.marked) &&
+ (n.right != null ==> n.right in S && n.right.marked));
+ {
+ var stack: seq<Node> := [];
+ call RecursiveMarkWorker(root, S, stack);
+ }
+
+ method RecursiveMarkWorker(root: Node, S: seq<Node>, stack: seq<Node>)
+ requires acc(S[*].marked, 50) && acc(S[*].*, 50);
+ requires root != null && root in S;
+ requires forall n in S :: n != null &&
+ ((n.left != null ==> n.left in S) &&
+ (n.right != null ==> n.right in S))
+ requires forall n in S :: n.marked ==>
+ (n in stack ||
+ ((n.left != null ==> n.left.marked) &&
+ (n.right != null ==> n.right.marked)));
+ requires forall n in stack :: n != null && n in S && n.marked;
+ ensures acc(S[*].marked, 50) && acc(S[*].*, 50);
+ ensures forall n in S :: n.left == old(n.left) && n.right == old(n.right);
+ ensures forall n in S :: n.marked ==>
+ (n in stack ||
+ ((n.left != null ==> n.left.marked) &&
+ (n.right != null ==> n.right.marked)));
+ ensures forall n in S :: old(n.marked) ==> n.marked;
+ ensures root.marked;
+ {
+ if (! root.marked) {
+ root.marked := true;
+ var next:seq<Node> := [root] ++ stack;
+ assert next[0] == root;
+ if (root.left != null) {
+ call RecursiveMarkWorker(root.left, S, next);
+ }
+
+ if (root.right != null) {
+ call RecursiveMarkWorker(root.right, S, next);
+ }
+ }
+ }
+
+ method IterativeMark(root: Node, S: seq<Node>)
+ requires acc(S[*].*);
+ requires root in S;
+ requires forall n in S :: n != null &&
+ (n.left != null ==> n.left in S) &&
+ (n.right != null ==> n.right in S);
+ requires forall n in S :: ! n.marked;
+ requires forall n in S :: ! n.l && ! n.r;
+ ensures acc(S[*].*);
+ ensures forall n in S :: n.left == old(n.left) && n.right == old(n.right);
+ ensures root.marked;
+ ensures forall n in S :: n.marked ==>
+ (n.left != null ==> n.left.marked) &&
+ (n.right != null ==> n.right.marked);
+ ensures forall n in S :: ! n.l && ! n.r;
+ {
+ var t:Node := root;
+ t.marked := true;
+ var stack: seq<Node> := [];
+
+ var stop := false;
+ while(!stop)
+ invariant acc(S[*].*);
+ invariant root.marked && t in S && t !in stack;
+ invariant forall n in stack :: n in S;
+ invariant forall i in [0..|stack|] :: forall j in [i+1..|stack|] :: stack[i] != stack[j];
+ invariant forall n in S :: (n in stack || n == t) ==>
+ n.marked &&
+ (n.r ==> n.l) &&
+ (n.l && n.left != null ==> n.left in S && n.left.marked) &&
+ (n.r && n.right != null ==> n.right in S && n.right.marked)
+ // stack is linked
+ invariant forall i in [1..|stack|] :: stack[i-1] == (stack[i].l ? stack[i].right : stack[i].left);
+ invariant 0 < |stack| ==> t == (stack[0].l ? stack[0].right : stack[0].left);
+ // goal
+ invariant forall n in S :: n.marked && n !in stack && n != t ==>
+ (n.left != null ==> n.left in S && n.left.marked) &&
+ (n.right != null ==> n.right in S && n.right.marked);
+ // preservation
+ invariant forall n in S :: n !in stack && n != t ==> ! n.l && ! n.r;
+ invariant forall n in S :: n.left == old(n.left) && n.right == old(n.right);
+ invariant stop ==> |stack| == 0 && ! t.l && ! t.r &&
+ (t.left != null ==> t.left.marked) &&
+ (t.right != null ==> t.right.marked);
+ {
+ if (! t.l && (t.left == null || t.left.marked)) {
+ // advance
+ t.l := true;
+ } else if (t.l && ! t.r && (t.right == null || t.right.marked)) {
+ // advance
+ t.r := true;
+ } else if (t.r) {
+ // pop
+ t.l := false;
+ t.r := false;
+ if (|stack| == 0) {
+ stop := true;
+ } else {
+ t := stack[0];
+ stack := stack[1..];
+ if (t.l) {t.r := true} else {t.l := true}
+ }
+ } else if (!t.l) {
+ // push
+ stack := [t] ++ stack;
+ assert stack[0] == t;
+ t := t.left;
+ t.marked := true;
+ } else if (!t.r) {
+ // push
+ assert t.l;
+ stack := [t] ++ stack;
+ assert stack[0] == t;
+ t := t.right;
+ t.marked := true;
+ }
+ }
+ }
+}