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class Node
{
var v:int;
var n:Node;
predicate inv
{ acc(v) && acc(n) && (n!=null ==> n.inv) }
function len():int
requires inv;
ensures result>0;
{
unfolding inv in (n==null) ? 1 : 1+n.len()
}
function get(i:int):int
requires inv && 0<=i && i<len();
{
unfolding inv in (i==0) ? v : n.get(i-1)
}
method addLast(x:int)
requires inv;
ensures inv;
ensures len()==old(len())+1 && get(old(len()))==x;
ensures (forall i:int :: 0<=i && i<old(len()) ==> get(i)==old(get(i)));
{
unfold inv;
if(n==null)
{
n:=new Node;
n.v:=x; n.n:=null;
fold n.inv;
}
else
{
call n.addLast(x);
}
fold inv;
}
method append(p:List)
requires inv && p!=null && p.inv;
ensures inv;
ensures len()==old(len()+p.len());
ensures (forall i in [0..old(len())] :: get(i)==old(get(i)));
ensures (forall i in [old(len())..len()] :: get(i)==old(p.get(i-len())));
{
unfold inv;
if(n==null)
{
unfold p.inv;
n:=p.c;
}
else
{
call n.append(p);
}
fold inv;
}
method remove(i:int)
requires inv && i>=0 && i<len()-1;
ensures inv;
ensures len()==old(len())-1;
ensures (forall j in [0..i+1] :: get(j)==old(get(j)));
ensures (forall j in [i+1..len()] :: get(j)==old(get(j+1)));
{
unfold inv;
if(i==0)
{
unfold n.inv;
n:=n.n;
}
else
{
call n.remove(i-1);
}
fold inv;
}
}
class List
{
var c:Node;
predicate inv { acc(c) && (c!=null ==> c.inv) }
function len():int
requires inv;
ensures result>=0;
{
unfolding inv in (c==null) ? 0 : c.len()
}
function get(i:int):int
requires inv && 0<=i && i<len();
{
unfolding inv in c.get(i)
}
method addFirst(x:int)
requires inv;
ensures inv;
ensures len()==old(len())+1 && get(0)==x;
ensures (forall i:int :: 1<=i && i<len() ==> get(i)==old(get(i-1)));
{
var p:Node;
unfold inv;
p:=new Node; p.v:=x; p.n:=c; c:=p;
fold c.inv;
assert c.len()==old(len())+1
fold inv;
}
method addLast(x:int)
requires inv;
ensures inv;
ensures len()==old(len())+1 && get(old(len()))==x;
ensures (forall i:int :: 0<=i && i<old(len()) ==> get(i)==old(get(i)));
{
unfold inv;
if(c==null)
{
c:=new Node;
c.v:=x; c.n:=null;
fold c.inv;
}
else
{
call c.addLast(x);
}
fold inv;
}
method append(p:List)
requires inv && p!=null && p.inv;
ensures inv;
ensures len()==old(len()+p.len());
ensures (forall i in [0..old(len())] :: get(i)==old(get(i)));
ensures (forall i in [old(len())..len()] :: get(i)==old(p.get(i-len())));
{
unfold inv;
if(c==null)
{
unfold p.inv;
c:=p.c;
}
else
{
call c.append(p);
}
fold inv;
}
method remove(i:int)
requires inv && i>=0 && i<len();
ensures inv;
ensures len()==old(len())-1;
ensures (forall j in [0..i] :: get(j)==old(get(j)));
ensures (forall j in [i..len()] :: get(j)==old(get(j+1)));
{
unfold inv;
if(i==0)
{
unfold c.inv;
c:=c.n;
}
else
{
call c.remove(i-1);
}
fold inv;
}
}
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