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# [Frama-c-discuss] question about a simple example and jessie

```> Another interesting
> tidbit is that Jessie is (probably rightly so) worried about integer
> overflows. Requiring an upper bound for tam seem to help prove some
> sub-properties.

My mistake: after fixing was was really wrong, the upper bound on tam
was not helping any more, so I removed it.

I changed the inner loop invariant to make it more readable to me, but
I don't know if the old one was wrong or not.

The way to prove the post-condition:

ensures \forall integer a; 0 <= a < tam
==> (\exists integer b; 0 <= b < tam
&& \old(vector[b]) == vector[a]) ;

is probably to prove that the property is preserved at each step of
the computation, that is, to include the same property in both the
inner and outer loop invariants. I named that property "SameElements".
I didn't get alt-ergo to prove either initialization (for the outer
loop!) or preservation (for the inner loop) of that invariant, but at
least the post-condition is proved from it.

Note that the specification is not a complete specification of a
sorting algorithm as one could expect: a function that copied
vector all over vector would satisfy it. But it would be a good
milestone to verify this specification before moving up to a complete
one, and a good second milestone on the way would be to add the
post-condition SameElements{Old, Here}(vector, tam) to the contract.

Frama-C 20100401 (Boron), Why 2.24 patched, alt-ergo 0.9, command-line:

frama-c -jessie b.c

_______

# pragma SeparationPolicy(none)

/*@ predicate Sorted{L}(int *a, integer l, integer h) =
\forall integer i; l <= i < h ==> a[i] <= a[i+1] ;
*/

/*@ predicate SameElements{Hr, Thr}(int *v, integer tam) =
\forall integer a; 0 <= a < tam
==> (\exists integer b; 0 <= b < tam
&& \at(v[b],Hr) == \at(v[a],Thr)) ;

*/

/*@
requires \valid(i) ;
requires \valid(j) ;
assigns *i, *j;
ensures *j == \old(*i) ;
ensures *i == \old(*j) ;
*/
void swap (int *i, int *j)
{
int tmp=*i;
*i = *j;
*j = tmp;
}

/*@
requires 0 < tam ;
requires \valid_range(vector, 0, tam-1) ;
assigns vector[0..tam-1] ;
ensures SameElements{Here, Old}(vector, tam) ;
ensures Sorted(vector, 0, tam-1) ;
*/
void bubbleSort(int *vector, int tam)
{
int j,i;
j = i = 0;

/*@
loop invariant 0 <= i <= tam ;
loop invariant Sorted{Here}(vector, tam-i-1, tam-1) ;
loop invariant 0 < i < tam
==> \forall int a, b; 0 <= b <= tam-i-1 <= a < tam
==> vector[a] >= vector[b] ;
loop invariant SameElements{Here, Pre}(vector, tam) ;
loop variant tam-i ;
*/
for (i=0; i<tam; i++)
{
/*@
loop invariant 0 <= j < tam-i ;
loop invariant
\forall int a; 0 <= a < j
==> vector[a] <= vector[j] ;
loop invariant SameElements{Here, Pre}(vector, tam) ;
loop variant tam-i-j-1 ;
*/
for (j=0; j<tam-i-1; j++)
{
if (vector[j] > vector[j+1])
swap(&vector[j], &vector[j+1]);
}
}
}

```