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## Hoare's Proof of a Program: FIND

Authors: Jean-Christophe Filliâtre

Tools: Why3

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

module FIND

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

constant _N: int (* actually N in Hoare's notation *)
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: int) (n: int) (i: int) (r: int) (a: array int) =
m <= i /\ (forall p:int. 1 <= p < i -> a[p] <= r) /\
(i <= n -> exists p:int. i <= p <= n /\ r <= a[p])

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

predicate termination (i:int) (j:int) (i0:int) (j0:int) (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 a (old a) }
'Init:
let m = ref 1 in let n = ref _N in
while !m < !n do
invariant { m_invariant !m a /\ n_invariant !n a /\
permut a (at a 'Init) /\ 1 <= !m /\ !n <= _N }
variant { !n - !m }
let r = a[f] in let i = ref !m in let j = ref !n in
while !i <= !j do
invariant { i_invariant !m !n !i r a /\ j_invariant !m !n !j r a /\
m_invariant !m a /\ n_invariant !n a /\ 0 <= !j /\ !i <= _N + 1 /\
termination !i !j !m !n r a /\ permut a (at a 'Init) }
variant { _N + 2 + !j - !i }
'L: while a[!i] < r do
invariant { i_invariant !m !n !i r a /\
at !i '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 r a /\
!j <= at !j '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 (at a 'L) !i !j };
assert { a[!i] <= r }; assert { r <= a[!j] };
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
```