Hoare's Proof of a Program: FIND

Authors: Jean-Christophe Filliâtre

Tools: Why3

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```(*
C. A. R. Hoare.
Proof of a program: Find.
Commun. ACM, 14:39--45, January 1971.
*)

module FIND

use int.Int
use ref.Ref
use array.Array
use array.ArrayPermut

val constant _N: int (* actually N in Hoare's notation *)
val constant f: int

axiom f_N_range: 1 <= f <= _N

predicate found (a: array int) =
forall p q:int. 1 <= p <= f <= q <= _N -> a[p] <= a[f] <= a[q]

predicate m_invariant (m: int) (a: array int) =
m <= f /\ forall p q:int. 1 <= p < m <= q <= _N -> a[p] <= a[q]

predicate n_invariant (n: int) (a: array int) =
f <= n /\ forall p q:int. 1 <= p <= n < q <= _N -> a[p] <= a[q]

predicate i_invariant (m n i q r: int) (a: array int) =
m <= i /\ (forall p:int. 1 <= p < i -> a[p] <= r) /\
(i <= n -> i <= q <= n /\ r <= a[q])

predicate j_invariant (m n j p r: int) (a: array int) =
j <= n /\ (forall q:int. j < q <= _N -> r <= a[q]) /\
(m <= j -> m <= p <= j /\ a[p] <= r)

predicate termination (i j i0 j0 r: int) (a:array int) =
(i > i0 /\ j < j0) \/ (i <= f <= j /\ a[f] = r)

let find (a: array int) =
requires { length a = _N+1 }
ensures  { found a /\ permut_all a (old a) }
let m = ref 1 in let n = ref _N in
while !m < !n do
invariant { m_invariant !m a /\ n_invariant !n a /\
permut_all a (old a) /\ 1 <= !m /\ !n <= _N }
variant { !n - !m }
let r = a[f] in let i = ref !m in let j = ref !n in
let ghost p = ref f in let ghost q = ref f in
while !i <= !j do
invariant { i_invariant !m !n !i !q r a /\ j_invariant !m !n !j !p r a /\
m_invariant !m a /\ n_invariant !n a /\ 0 <= !j /\ !i <= _N + 1 /\
termination !i !j !m !n r a /\ permut_all a (old a) }
variant { _N + 2 + !j - !i }
label L in
while a[!i] < r do
invariant { i_invariant !m !n !i !q r a /\
!i at L <= !i <= !n /\ termination !i !j !m !n r a }
variant { _N + 1 - !i }
i := !i + 1
done;

while r < a[!j] do
invariant { j_invariant !m !n !j !p r a /\
!j <= !j at L /\ !m <= !j /\ termination !i !j !m !n r a }
variant { !j }
j := !j - 1
done;

assert { a[!j] <= r <= a[!i] };

if !i <= !j then begin
let w = a[!i] in begin a[!i] <- a[!j]; a[!j] <- w end;
assert { exchange a (a at L) !i !j };
ghost begin
p := if !i < !j then !i else !j - 1;
q := if !i < !j then !j else !i + 1
end;
i := !i + 1;
j := !j - 1
end
done;

assert { !m < !i /\ !j < !n };

if f <= !j then
n := !j
else if !i <= f then
m := !i
else
begin n := f; m := f end
done

end
```