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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)


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(* The 2nd Verified Software Competition (VSTTE 2012)
   https://sites.google.com/site/vstte2012/compet

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

module TwoWaySort

  use int.Int
  use bool.Bool
  use ref.Refint
  use array.Array
  use array.ArraySwap
  use 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 }
  = 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 (old a) 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

ObligationsAlt-Ergo 1.30
VC two_way_sort0.09