Programs with Proofs: Finding Maximum in an Array

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

Finding Maximum in an Array


/* Method findmax() returns the index of the maximum element in array a[] of
   n elements */

typedef unsigned int uint;

/*@
  // is the element a[m] the maximum among indices 0 to e?
  predicate is_max(int *a, integer e, integer m) =
    \forall integer i; 0 <= i <= e ==> a[i] <= a[m];
 */

/*@
  requires n > 0;
  requires \valid_read(a + (0..n-1));
  assigns \nothing;
  ensures is_max(a, n-1, \result);
 */
uint findmax(int a[], uint n)
{
  uint i, m;

  i = 1;
  m = 0;

  /*@
    loop assigns i, m;
    loop invariant is_max(a, i-1, m);
    loop invariant 0 <= m <= n-1;
    loop variant n-i;
   */
  while(i < n)
  {
    if(a[i] > a[m])
      m = i;

    i++;
  }

  return m;
}