## Dijkstra's national flag (variant)

Variant with number of occurrences instead of predicate permut

Auteurs: Jean-Christophe Filliâtre

Catégories: Array Data Structure

Outils: Why3

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

# Dijkstra's "Dutch national flag"

Variant with number of occurrences instead of predicate [permut]

```module Flag

use int.Int
use map.Map
use ref.Ref

type color = Blue | White | Red

let eq_color (c1 c2 :color) : bool
ensures { result <-> c1 = c2 }
= match c1,c2 with
| Blue,Blue | Red,Red | White,White -> True
| _,_ -> False
end

predicate monochrome (a: map int color) (i: int) (j: int) (c: color) =
forall k: int. i <= k < j -> a[k]=c

let rec function nb_occ (a: map int color) (i: int) (j: int) (c: color) : int
variant { j - i }
= if i >= j then 0 else
if eq_color a[j-1] c then 1 + nb_occ a i (j-1) c else nb_occ a i (j-1) c

let rec lemma nb_occ_split (a: map int color) (i j k: int) (c: color)
requires { i <= j <= k }
variant { k - j }
ensures { nb_occ a i k c = nb_occ a i j c + nb_occ a j k c }
= if k = j then () else nb_occ_split a i j (k-1) c

let rec lemma nb_occ_ext (a1 a2: map int color) (i j: int) (c: color)
requires { forall k: int. i <= k < j -> a1[k] = a2[k] }
variant { j - i }
ensures { nb_occ a1 i j c = nb_occ a2 i j c }
= if i >= j then () else nb_occ_ext a1 a2 i (j-1) c

lemma nb_occ_store_outside_up:
forall a: map int color, i j k: int, c: color.
i <= j <= k -> nb_occ (set a k c) i j c = nb_occ a i j c

lemma nb_occ_store_outside_down:
forall a: map int color, i j k: int, c: color.
k < i <= j -> nb_occ (set a k c) i j c = nb_occ a i j c

lemma nb_occ_store_eq_eq:
forall a: map int color, i j k: int, c: color.
i <= k < j -> a[k] = c ->
nb_occ (set a k c) i j c = nb_occ a i j c

let rec lemma nb_occ_store_eq_neq (a: map int color) (i j k: int) (c: color)
requires { i <= k < j }
requires { a[k] <> c }
variant  { j - k }
ensures  { nb_occ (set a k c) i j c = nb_occ a i j c + 1 }
= if k = j - 1 then () else nb_occ_store_eq_neq a i (j-1) k c

let lemma nb_occ_store_neq_eq
(a: map int color) (i j k: int) (c c': color)
requires { i <= k < j } requires { c <> c' } requires { a[k] = c }
ensures  { nb_occ (set a k c') i j c = nb_occ a i j c - 1 }
= nb_occ_split a i k j c; nb_occ_split (set a k c') i k j c;
nb_occ_split a k (k + 1) j c; nb_occ_split (set a k c') k (k+1) j c

let lemma nb_occ_store_neq_neq
(a: map int color) (i j k: int) (c c': color)
requires { i <= k < j } requires { c <> c' } requires { a[k] <> c }
ensures  { nb_occ (set a k c') i j c = nb_occ a i j c }
= nb_occ_split a i k j c; nb_occ_split (set a k c') i k j c;
nb_occ_split a k (k + 1) j c; nb_occ_split (set a k c') k (k+1) j c

use array.Array

let swap (a:array color) (i: int) (j: int) : unit
requires { 0 <= i < a.length }
requires { 0 <= j < a.length }
ensures { a[i] = old a[j] }
ensures { a[j] = old a[i] }
ensures { forall k: int. k <> i /\ k <> j -> a[k] = old a[k] }
ensures { forall k1 k2: int, c: color. k1 <= i < k2 /\ k1 <= j < k2 ->
nb_occ a.elts k1 k2 c = nb_occ (old a.elts) k1 k2 c }
= let ai = a[i] in
let aj = a[j] in
a[i] <- aj;
a[j] <- ai

let dutch_flag (a:array color)
ensures { (exists b: int. exists r: int.
monochrome a.elts 0 b Blue /\
monochrome a.elts b r White /\
monochrome a.elts r a.length Red) }
ensures { forall c: color.
nb_occ a.elts 0 a.length c = nb_occ (old a.elts) 0 a.length c }
= let b = ref 0 in
let i = ref 0 in
let r = ref a.length in
while !i < !r do
invariant { 0 <= !b <= !i <= !r <= a.length }
invariant { monochrome a.elts 0 !b Blue }
invariant { monochrome a.elts !b !i White }
invariant { monochrome a.elts !r a.length  Red }
invariant {
forall c: color.
nb_occ a.elts 0 a.length c = nb_occ (old a.elts) 0 a.length c }
variant { !r - !i }
match a[!i] with
| Blue -> swap a !b !i; b := !b + 1; i := !i + 1
| White -> i := !i + 1
| Red -> r := !r - 1; swap a !r !i
end
done

end
```