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# [Frama-c-discuss] 2d array and \separated with WP

• Subject: [Frama-c-discuss] 2d array and \separated with WP
• From: siegel at udel.edu (Stephen Siegel)
• Date: Wed, 16 May 2018 05:07:00 -0400

```Hello Frama-C list,
Iâ€™m trying to prove (using WP) some simple programs dealing with 2-dimensional arrays, represented as arrays of pointers.   Iâ€™m running into trouble reasoning about separation.  Can anyone tell me if the following lemma is valid?   It may be that I am misunderstanding the meaning of \separated.  In any case, I havenâ€™t been able to prove this lemmaâ€¦

/*@
predicate Sep2d(int **a, int n, int m) =
\separated(a[0..n-1]+(0..m-1));

predicate SepRows(int **a, int n, int m) =
\forall integer i1, i2, j1, j2;
(0<=i1<n && 0<=i2<n && i1!=i2 && 0<=j1<m && 0<=j2<m) ==>
\separated(&a[i1][j1], &a[i2][j2]);

lemma SepLem1: \forall int **a, int n, int m;
(n>=0 && m>=0 && Sep2d(a, n, m)) ==> SepRows(a, n, m);

*/

One thing I noticed, using the Frama-C GUI and Why3 as prover, is that my Sep2d predicate seems to have been translated to â€œtrueâ€.

Below is the actual function I am trying to prove, which copies a 2d array:

/*@
predicate EqualRange(int *a, integer n, int *b) =
\forall integer i; 0<=i<n ==> a[i] == b[i];

predicate EqualRange2d(int **a, integer n, integer m, int **b) =
\forall integer i; 0<=i<n ==> EqualRange(a[i], m, b[i]);

predicate Valid2d(int **a, integer n, integer m) =
\valid(a + (0..n-1)) && \valid(a[0..n-1] + (0..m-1));
*/

/*@ requires n>=0 && m>=0;
@ requires Valid2d(a, n, m);
@ requires Valid2d(b, n, m);
@ requires \separated(a + (0..n-1), a[0..n-1] + (0..m-1), b + (0..n-1), b[0..n-1] + (0..m-1), &n, &m);
@ assigns a[0..n-1][0..m-1];
@ ensures EqualRange2d(a, n, m, b);
@*/
void copy2d(int **a, int n, int m, int **b) {
/*@ loop invariant i1: 0<=i<=n;
@ loop invariant i2: EqualRange2d(a, i, m, b);
@ loop assigns i, a[0..n-1][0..m-1];
@ loop variant n - i;
@*/
for (int i=0; i<n; i++) {
/*@ loop invariant 0<=j<=m;
@ loop invariant z1: EqualRange2d(a, i, m, b);
@ loop invariant z2: EqualRange(a[i], j, b[i]);
@ loop assigns j, a[i][0..m-1];
@ loop variant m - j;
@*/
for (int j=0; j<m; j++) {
// I canâ€™t prove the following assertion â€¦
//@ assert sep1 : \separated(&a[i][j], a[0..i-1] + (0..m-1));
a[i][j] = b[i][j];
}
}
}

-Steve

```