Programs with Proofs: Selection Sort

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

Selection Sort


/* Method selection_sort() sorts the array a[] of n elements using
   Selection Sort algorithm. */

/*@
  predicate sorted(int *a, integer n) =
    \forall integer i; 0 <= i <= n-2 ==> a[i] <= a[i+1];

  predicate swap_at_indices{L1, L2}(int *a, integer i, integer j) =
    \at(a[i], L1) == \at(a[j], L2) && \at(a[i], L2) == \at(a[j], L1);

  // array a[] (of n elements) at labels L1 and L2 differs only by a swap at indices
  // i and j
  predicate swap_in_array{L1, L2}(int *a, integer n, integer i, integer j) =
    swap_at_indices{L1, L2}(a, i, j) &&
    \forall integer k; (0 <= k <= n-1 && k!=i && k!=j) ==>
    \at(a[k], L1) == \at(a[k], L2);

  // define how array a[] (of n elements) at label L1 is a permutation of it at
  // label L2
  inductive permutation{L1, L2}(int *a, integer n)
  {
    case reflexive{L1}:
      \forall int *a, integer n; permutation{L1, L1}(a, n);
      
    case swap{L1, L2}:
      \forall int *a, integer n, i, j; 0 <= i < j <= n-1 &&
      swap_in_array{L1, L2}(a, n, i, j) ==> permutation{L1, L2}(a, n);

    case transitive{L1, L2, L3}:
      \forall int *a, integer n; permutation{L1, L2}(a, n) &&
      permutation{L2, L3}(a, n) ==> permutation{L1, L3}(a, n);
  }
 */

/*@
  requires \valid(a+i) && \valid(a+j);
  assigns a[i], a[j];
  ensures swap_at_indices{Pre, Post}(a, i, j);
 */
static void swap(int a[], int i, int j)
{
  int t;
  t = a[i];
  a[i] = a[j];
  a[j] = t;
}

/*@
  requires n >= 1;
  requires \valid(a + (0..n-1));
  assigns a[0..n-1];
  ensures sorted(a, n);
  ensures permutation{Pre, Post}(a, n);
 */
void selection_sort(int a[], int n)
{
  int i;

  i = 0;

  /*@
    loop assigns i, a[0..n-1];
    loop invariant P: \forall integer k; i-1 >= 0 && i <= k <= n-1 ==>
                      a[i-1] <= a[k];
    loop invariant Q: sorted(a, i);
    loop invariant 0 <= i <= n-1;
    loop invariant permutation{Pre, LoopCurrent}(a, n);
    loop variant n-i;
   */
  while(i < n-1)
  {
    int min = i;
    int j = i + 1;

    /*@
      loop assigns min, j;
      loop invariant \forall integer k; i <= k <= j-1 ==> a[min] <= a[k];
      loop invariant i+1 <= j <= n;
      loop invariant i <= min <= n-1;
      loop variant n-j;
     */
    while(j < n)
    {
      if(a[j] < a[min])
        min = j;

      j = j + 1;
    }

    if(min != i)
    {
      swap(a, i, min);
      //@assert swap_in_array{LoopCurrent, Here}(a, n, i, min);
    }

    i++;
  }

  /* P and Q hold with (i = n-1) */
  /* Now a[n-1] must be among the largest, so a[0:n-1] is sorted */
}