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class AVLTree{
var root : AVLTreeNode;
predicate valid{
acc(root,100)
&& (root!=null ==> root.valid)
&& (root!=null ==> acc(root.parent,100))
&& (root!=null ==> root.parent==null)
&& (root!=null ==> acc(root.root,50))
&& (root!=null ==> root.root==root)
}
method init()
requires acc(root,100);
ensures valid;
{
root := null;
fold valid;
}
method has(k : int) returns (b : bool)
requires valid;
ensures valid;
{
unfold valid;
if (root==null){
b := false;
fold valid;
}else{
var n : AVLTreeNode := root;
b := false;
var end : bool := false;
fold n.udParentValid;
while (!end)
invariant acc(root,100);
invariant root != null && acc(root.parent,50);
invariant n!=null;
invariant n.valid;
invariant acc(n.root,40);
invariant n.udParentValid;
invariant unfolding n.valid in n.root==root;
invariant root!=null;
{
unfold n.valid;
unfold n.validRest;
if (n.key==k){
b := true;
fold n.validRest;
fold n.valid;
end := true;
}else{
if (n.key<k){
if (n.left==null){
end := true;
fold n.validRest;
fold n.valid;
}else{
var p : AVLTreeNode := n;
unfold p.leftValid;
n := p.left;
p.leftDown := true;
fold p.leftOpen;
fold p.udValid;
assert p.right!=p.left;
assert n.parent.left==n;
fold n.udParentValid;
}
}else{
if (n.right==null){
end := true;
fold n.validRest;
fold n.valid;
}else{
var p : AVLTreeNode := n;
unfold p.rightValid;
n := p.right;
p.leftDown := false;
fold p.rightOpen;
fold p.udValid;
fold n.udParentValid;
}
}
}
}
end := false;
while (!end)
invariant acc(root,100);
invariant root != null && acc(root.parent,50);
invariant n!=null;
invariant n.valid;
invariant n.udParentValid;
invariant acc(n.root,40);
invariant unfolding n.valid in n.root==root;
invariant root!=null;
invariant end==>unfolding n.udParentValid in n.parent==null;
{
unfold n.udParentValid;
var p : AVLTreeNode := n.parent;
if (p==null){
end := true;
fold n.udParentValid;
}else{
unfold p.udValid;
if (p.left==n){
unfold p.leftOpen;
fold p.leftValid;
}else{
unfold p.rightOpen;
fold p.rightValid;
}
fold p.validRest;
fold p.valid;
n:=p;
}
}
assert unfolding n.udParentValid in n==root;
assert acc(n.root,40);
unfold n.udParentValid;
assert acc(n.root,50);
fold valid;
}
}
}
class AVLTreeNode{
var key : int;
var left : AVLTreeNode;
var right : AVLTreeNode;
var parent : AVLTreeNode;
ghost var leftDown : bool;
ghost var root : AVLTreeNode;
predicate valid{
validRest
&& leftValid
&& rightValid
}
predicate validRest{
acc(key ,100)
&& acc(root, 30)
&& acc(left ,75)
&& acc(right ,75)
&& acc(leftDown,100)
&& (right!=left || right==null)
}
predicate rightValid{
acc(right ,25)
&& acc(root,10)
&& (right!=null ==> right.valid)
&& (right!=null ==> acc(right.parent,100))
&& (right!=null ==> right.parent==this)
&& (right!=null ==> acc(right.root,50))
&& (right!=null ==> right.root==root)
}
predicate leftValid{
acc(left ,25)
&& acc(root,10)
&& (left!=null ==> left.valid)
&& (left!=null ==> acc(left.parent,100))
&& (left!=null ==> left.parent == this)
&& (left!=null ==> acc(left.root,50))
&& (left!=null ==> left.root == root)
}
predicate leftOpen{
acc(left ,25)
&& acc(root,10)
&& (left!=null ==> acc(left.parent,50))
&& (left!=null ==> left.parent==this)
}
predicate rightOpen{
acc(right ,25)
&& acc(root,10)
&& (right!=null ==> acc(right.parent,50))
&& (right!=null ==> right.parent==this)
}
predicate udParentValid {
acc(parent,50)
&& acc(root,10)
&& (parent!=null ==> parent.udValid)
&& (parent!=null ==> acc(parent.leftDown,50))
&& (parent!=null ==> acc(parent.left,50))
&& (parent!=null ==> ( parent.leftDown<==>parent.left==this))
&& (parent!=null ==> acc(parent.right,50))
&& (parent!=null ==> (!parent.leftDown<==>parent.right==this))
&& (parent!=null ==> acc(parent.root,50))
&& (parent!=null ==> root==parent.root)
&& (parent==null ==> root==this)
}
predicate udValid{
acc(key ,100)
&& acc(leftDown,50)
&& acc(left ,25)
&& acc(right ,25)
&& acc(root ,20)
&& ( leftDown ==> rightValid)
&& ( leftDown ==> leftOpen )
&& (!leftDown ==> leftValid )
&& (!leftDown ==> rightOpen )
&& udParentValid
}
method init(k : int)
requires acc(key ,100);
requires acc(left ,100);
requires acc(right ,100);
requires acc(leftDown ,100);
requires acc(root, 100);
ensures valid;
{
left := null;
right := null;
key := k;
fold leftValid;
fold rightValid;
fold validRest;
fold valid;
}
}
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