## Mergesort (queues)

Sorting queues using mergesort.

Auteurs: Jean-Christophe Filliâtre

Catégories: Sorting Algorithms

Outils: Why3

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# Sorting a queue using mergesort

Author: Jean-Christophe FilliĆ¢tre (CNRS)

```module MergesortQueue

use int.Int
use seq.Seq
use seq.Mem
use seq.FreeMonoid
use seq.Occ
use seq.Permut
use queue.Queue

type elt
val predicate le elt elt
clone relations.TotalPreOrder with
type t = elt, predicate rel = le, axiom .
clone export seq.Sorted with
type t = elt, predicate le = le, goal .

let merge (q1: t elt) (q2: t elt) (q: t elt)
requires { q = empty /\ sorted q1 /\ sorted q2 }
ensures  { sorted q }
ensures  { length q = length (old q1) + length (old q2) }
ensures  { forall x. occ_all x q = occ_all x (old q1) + occ_all x (old q2) }
= while not (is_empty q1 && is_empty q2) do
invariant { sorted q1 /\ sorted q2 /\ sorted q }
invariant { forall x y: elt. mem x q -> mem y q1 -> le x y }
invariant { forall x y: elt. mem x q -> mem y q2 -> le x y }
invariant { length q + length q1 + length q2 = length (old q1) + length (old q2) }
invariant { forall x. occ_all x q + occ_all x q1 + occ_all x q2
= occ_all x (old q1) + occ_all x (old q2) }
variant   { length q1 + length q2 }
if is_empty q1 then
push (safe_pop q2) q
else if is_empty q2 then
push (safe_pop q1) q
else
let x1 = safe_peek q1 in
let x2 = safe_peek q2 in
if le x1 x2 then
push (safe_pop q1) q
else
push (safe_pop q2) q
done

let rec mergesort (q: t elt) : unit
ensures { sorted q /\ permut_all q (old q) }
variant { length q }
= if Peano.gt (length q) Peano.one then begin
let q1 = create () : t elt in
let q2 = create () : t elt in
while not (is_empty q) do
invariant { forall x. occ_all x q1 + occ_all x q2 + occ_all x q = occ_all x (old q) }
invariant { length (old q) = length q1 + length q2 + length q }
invariant { length q1 = length q2 \/
length q = 0 /\ length q1 = length q2 + 1 }
variant   { length q }
let x = safe_pop q in push x q1;
if not (is_empty q) then let x = safe_pop q in push x q2
done;
assert { q = empty };
assert { forall x. occ_all x q1 + occ_all x q2 = occ_all x (old q) };
mergesort q1;
mergesort q2;
merge q1 q2 q
end else
assert { q = empty \/ exists x: elt. q = cons x empty }

end
```