Programs with Proofs: Array Copy with Overlap

Method of Proving

This is a program which also contains annotations to help prove its correctness using the Frama-C software.

The annotations specify various properties for a method like the pre-condition, post-condition, assertions, states modified, loop-invariants and loop-variants. They are also used to define other artifacts to ease specification/proving like predicates, logic functions, axioms and lemmas. These annotations are provided inside code-comments of the form /*@ ... */ and //@ ....

Frama-C with its WP plugin works with external provers like Alt-Ergo and CVC4 to automatically prove these specified properties. The WP plugin internally works based on the Weakest Precondition calculus. You can read more about this plugin's usage and meaning of the annotations in this tutorial.

The programs provided here have been proved using these versions of the tools: Frama-C (contains WP plugin) 21.1, Alt-Ergo 2.3.1 and CVC4 1.6. The system is x86_64 running Debian 10 Linux.

In addition to proving the specified properties, we will also be checking for other issues like overflows via another Frama-C plugin called RTE (by using option -wp-rte).

Command to prove this program:
frama-c -wp -wp-prover cvc4 -wp-rte filename.c

Array Copy with Overlap


/* Method copy() copies n chars from source array s[] to destination array d[];
   the two arrays (of n elements each) are allowed to overlap.
   Note that the implementation uses a pointer-comparison (d < s) which is
   undefined in the C language.
 */

/*@
  requires n >= 0;
  requires \valid(d + (0..n-1)) && \valid(s + (0..n-1));
  assigns d[0..n-1];
  ensures \forall int i; 0 <= i <= n-1 ==> d[i] == \old(s[i]);
 */
void copy(char d[], char s[], int n)
{
  int i;

  if(d == s || n == 0)
    return;

  if(d < s)
  {
    /* Either d[] and s[] do not overlap, or the overlap is after some chars
       of d[]. Copy in forward direction. */

    i = 0;

    /*@
      loop assigns d[0..n-1], i;
      loop invariant 0 <= i <= n;
      loop invariant \forall int j; 0 <= j <= i-1 ==> d[j] == \at(s[j], Pre);

      // Total i chars written in d[], and upto i-1 chars written at the beginning
      // of s[] due to possible overlap.
      loop invariant \forall int j; i-1 <= j <= n-1 ==> s[j] == \at(s[j], Pre);
      loop variant n-i;
     */
    while(i < n)
    {
      d[i] = s[i];
      i = i + 1;
    }
  }
  else
  {
    /* d > s */
    /* Either d[] and s[] do not overlap, or the overlap is after some chars
       of s[]. Copy in backward direction. */

    i = n-1;

    /*@
      loop assigns d[0..n-1], i;
      loop invariant -1 <= i <= n-1;
      loop invariant \forall int j; i+1 <= j <= n-1 ==> d[j] == \at(s[j], Pre);

      // Total n-1-i chars written in d[], and upto n-2-i chars written at the end
      // of s[] due to possible overlap.
      loop invariant \forall int j; 0 <= j <= i+1 ==> s[j] == \at(s[j], Pre);
      loop variant i+1;
     */
    while(i >= 0)
    {
      d[i] = s[i];
      i = i - 1;
    }
  }
}