## Sort an array of Boolean values

Problem 1 of the second verified software competition.

The ZIP file below contains both the source code, the Why3 proof session file and the Coq script of the proof made in Coq. The Why3 source code is then displayed, followed by a summary of the proofs.

Authors: Jean-Christophe Filliâtre

Topics: Array Data Structure

Tools: Why3

References: The 2nd Verified Software Competition

see also the index (by topic, by tool, by reference, by year)

```(* The 2nd Verified Software Competition (VSTTE 2012)

Problem 1:
Sorting an array of Boolean values, using swaps only *)

module TwoWaySort

use import int.Int
use import bool.Bool
use import ref.Refint
use import array.Array
use import array.ArraySwap
use import array.ArrayPermut

predicate le (x y: bool) = x = False \/ y = True

predicate sorted (a: array bool) =
forall i1 i2: int. 0 <= i1 <= i2 < a.length -> le a[i1] a[i2]

let two_way_sort (a: array bool)
ensures { sorted a }
ensures { permut_all (old a) a }
= 'Init:
let i = ref 0 in
let j = ref (length a - 1) in
while !i < !j do
invariant { 0 <= !i /\ !j < length a }
invariant { forall k: int. 0 <= k < !i -> a[k] = False }
invariant { forall k: int. !j < k < length a -> a[k] = True }
invariant { permut_all (at a 'Init) a }
variant   { !j - !i }
if not a[!i] then incr i
else if a[!j] then decr j
else begin swap a !i !j; incr i; decr j end
done

end
```

# Why3 Proof Results for Project "vstte12_two_way_sort"

## Theory "vstte12_two_way_sort.TwoWaySort": fully verified in 0.00 s

 Obligations Alt-Ergo (0.99.1) CVC3 (2.4.1) 1. VC for two_way_sort --- --- split_goal_wp 1. loop invariant init 0.02 --- 2. loop invariant init 0.02 --- 3. loop invariant init 0.01 --- 4. loop invariant init 0.02 --- 5. type invariant 0.01 --- 6. index in array bounds 0.02 --- 7. loop invariant preservation 0.03 --- 8. loop invariant preservation 0.02 --- 9. loop invariant preservation 0.02 --- 10. loop invariant preservation 0.02 --- 11. loop variant decrease 0.02 --- 12. index in array bounds 0.01 --- 13. loop invariant preservation 0.02 --- 14. loop invariant preservation 0.02 --- 15. loop invariant preservation 0.02 --- 16. loop invariant preservation 0.02 --- 17. loop variant decrease 0.01 --- 18. precondition 0.02 --- 19. loop invariant preservation 0.04 --- 20. loop invariant preservation 0.04 --- 21. loop invariant preservation --- --- split_goal_wp 1. loop invariant preservation --- 0.53 22. loop invariant preservation 0.10 --- 23. loop variant decrease 0.02 --- 24. type invariant 0.01 --- 25. postcondition 0.02 --- 26. postcondition 0.02 ---