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

Lemma sqrt_step_correct rec i j:
0 < φ i -> 0 < φ j -> φ i < (φ j + 1) ^ 2 ->
2 * φ j < wB ->
(forall j1 : int,
0 < φ j1 < φ j -> φ i < (φ j1 + 1) ^ 2 ->
φ (rec i j1) ^ 2 <= φ i < (φ (rec i j1) + 1) ^ 2) ->
φ (sqrt_step rec i j) ^ 2 <= φ i < (φ (sqrt_step rec i j) + 1) ^ 2.