(* Sqrt *)
(* Direct transcription of an old proof of a fortran program in boyer-moore *)

Lemma sqrt2_step_correct rec ih il j:
2 ^ (Z_of_nat (size - 2)) <= φ ih ->
0 < φ j -> Φ (WW ih il) < (φ j + 1) ^ 2 ->
(forall j1, 0 < φ j1 < φ j ->  Φ (WW ih il) < (φ j1 + 1) ^ 2 ->
φ (rec ih il j1) ^ 2 <= Φ (WW ih il) < (φ (rec ih il j1) + 1) ^ 2) ->
φ (sqrt2_step rec ih il j) ^ 2 <= Φ (WW ih il)
< (φ (sqrt2_step rec ih il j) + 1) ^  2.