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/**
Interesting issues:
* using functions in refinement to write getters
* using functions to write coupling invariant
* refining to a recursive implementation
* restricting refinement (List1 must have at least one item)
* refining a method with an output variable
Do we shadows the abstract variables in the later concrete programs?
How do we handle generic sequence in List2
*/
class List0 {
var rep: seq<int>;
method init()
requires acc(this.*);
ensures acc(rep);
{
rep := [0];
}
function length(): int
requires acc(rep);
{
|rep|
}
method get(i: int) returns (v: int)
requires acc(rep);
requires 0 <= i && i < length();
ensures acc(rep);
{
v := rep[i];
}
method pick() returns (v: int)
requires acc(rep);
ensures acc(rep);
{
var i: int;
assume 0 <= i && i < length();
v := rep[i];
}
}
class List1 {
ghost var rep: seq<int>;
var data: int;
var l: seq<List1>;
function inv(): bool
requires acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l);
{
/** valid */ |l| >= 0 &&
(forall i in [0..|l|] :: l[i] != null && l[i].l == l[i+1..]) &&
/** coupling */ |l| + 1 == |rep| &&
(forall i in [0..|l|] :: l[i].data == rep[i+1]) &&
data == rep[0]
}
method init()
requires acc(this.*);
ensures acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && inv();
{
rep := [0];
data := 0;
l := nil<List1>;
}
function length(): int
requires acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && inv();
{
|l| + 1
}
method checkLength()
requires acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && inv();
{
assert |l| + 1 == |rep|;
}
method get(i: int) returns (v: int)
requires acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && inv();
requires 0 <= i && i < length();
ensures acc(rep) && acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && inv();
{
if (i == 0) {
v := data;
} else {
v := l[i-1].data;
}
assert v == rep[i];
}
}
class List2 {
// ghost var rep: seq<int>;
ghost var l: seq<List2>;
var data: int;
var next: List2;
var size: int;
function inv(): bool
requires acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next);
{
/** valid */ |l| >= 0 &&
(forall i in [0..|l|] :: l[i] != null && l[i].l == l[i+1..]) &&
/** new coupling */ size == |l| + 1 &&
(next == null ==> |l| == 0) &&
(next != null ==> |l| > 0 && next == l[0] && l[|l|-1].next == null) &&
(forall i in [0..|l|] :: l[i].size == size - i - 1) &&
(forall i in [0..|l|-1] :: l[i].next == l[i+1])
}
method init()
requires acc(this.*);
ensures acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next) && inv();
{
data := 0;
l := nil<List2>;
next := null;
size := 1;
}
function length(): int
requires acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next) && inv();
{
size
}
method checkLength()
requires acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next) && inv();
{
assert size == |l| + 1;
}
method get(i: int) returns (v: int)
requires acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next) && inv();
requires 0 <= i && i < length();
ensures acc(l) && acc(data) && acc(l[*].data) && acc(l[*].l) && acc(size) && acc(next) && acc(l[*].size) && acc(l[*].next) && inv();
/** loop invariant: assertion on coupling of abstract and concrete outputs */
ensures i == 0 ==> v == data;
ensures i > 0 ==> i-1 < |l| && l[i-1] != null && v == l[i-1].data;
{
if (i == 0) {
v := data;
} else {
assert next != null;
assert l == [next] ++ next.l;
call w := next.get(i-1);
v := w;
}
}
}
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