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# [Frama-c-discuss] several WP questions

• Subject: [Frama-c-discuss] several WP questions
• From: MarkoSchuetz at web.de (Marko Schütz Schmuck)
• Date: Thu, 12 Feb 2015 13:48:32 -0400

```Dear All,

verifying an implementation of partition I have something like

/*@ axiomatic List {
@ type listInt;
@ logic listInt nil;
@ logic listInt cons(integer x, listInt xs);
@ logic listInt append(listInt xs, listInt ys);
@ logic listInt arrayToList{L}(int *arr, integer length);
@ logic boolean member(integer elem, listInt xs);
@ axiom appendNil:
@   \forall listInt ys; append(nil, ys) == ys;
@ axiom appendCons:
@   \forall listInt xs, ys; \forall integer x; append(cons(x, xs), ys) == cons(x, append(xs, ys));
@ axiom arrayToListNull{L}:
@   \forall int *arr; \forall integer i; i == 0 ==> arrayToList{L}(arr, i) == nil;
@ axiom arrayToListN{L}:
@   \forall int *arr; \forall integer length, newLength; length > 0 && newLength == length-1
@   ==> arrayToList{L}(arr, length) == cons(\at(arr, L), arrayToList{L}(arr+1, newLength));
@ axiom memberNil:
@   \forall integer elem; !member(elem, nil);
@   \forall integer elem; \forall listInt xs; member(elem, cons(elem, xs));
@ axiom memberConsTail:
@   \forall integer elem, x; \forall listInt xs; member(elem, xs) ==> member(elem, cons(x, xs));
@ predicate permutationLists(listInt a, listInt b);
@ axiom permutationListsNil:
@   permutationLists(nil, nil);
@ axiom permutationListsCons:
@   \forall listInt a1, a2, a3, ta, b1, b2, b3, tb; ta == append(a1, append(a2, a3))
@   && tb == append(b1, append(b2, b3)) && a2 != nil && a2 == b2
@   && permutationLists(append(a1, a3), append(b1, b3))
@   ==> permutationLists(ta, b1);
@}
@*/

/*@
@predicate permutation{L1, L2}(int *a, int *b, integer count) =
@  permutationLists(arrayToList{L1}(a, count), arrayToList{L2}(b, count));
@
@predicate property(integer x);
@*/

/*@ assigns \nothing;
@ ensures \result == \true <==> property(x);
@*/
int property(int x);

/*@ requires length >= 0 && \valid(arr+(0..length-1)) && \valid(ind+(0..length-1));
@ ensures 0 <= \result < length;
@ ensures \forall int i; 0 <= i <= \result ==> property(arr[ind[i]]);
@ ensures \forall int i; \result + 1 <= i < length ==> !property(arr[ind[i]]);
@ assigns ind[0..length-1];
@*/
int partition(int *arr, int *ind, int length) {
int gr = 0, j = length-1;
/*@ loop invariant 0 <= gr <= j+1 <= length ;
@ loop invariant \forall integer i; 0 <= i < gr ==> property(arr[ind[i]]);
@ loop invariant \forall integer i; j+1 <= i < length ==> !property(arr[ind[i]]);
@ loop invariant permutationLists(append(arrayToList{Here}(ind, gr), arrayToList{Here}(ind+j+1, length-j)), arrayToList{Here}(arr, gr + length - j));
@ loop variant j - gr;
@*/
while (gr <= j) {
if (property(arr[gr + length - 1 - j])) {
ind[gr] = gr + length - 1 - j;
//@ assert ind[gr] == gr + length - 1 - j && property(arr[gr + length - 1 - j]) && property(arr[ind[gr]]);
gr++;
} else {
ind[j] = gr + length - 1 - j;
j--;
}
}
return gr-1;
}

Frama-C Neon/WP failed to discharge the second part of the invariant,
so I started experimenting with some assertions. When I write the
assertion as above the PO is not discharged. When I turn on 'split'
the first of the three generated POs is discharged. When I write the
three assertions as separate assertions:

//@ assert ind[gr] == gr + length - 1 - j;
//@ assert property(arr[gr + length - 1 - j]);
//@ assert property(arr[ind[gr]]);

then the first and the last are discharged.

Any hints on how I should deal with this and how I could get the
invariant proved?

Thanks and best regards,

Marko
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