(* Sqrt *)

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

Ltac elim_div :=
  unfold Z.div, Z.modulo;
  match goal with
  |  H : context[ Z.div_eucl ?X ?Y ] |-  _ =>
     generalize dependent H; generalize (Z_div_mod_full X Y) ; case (Z.div_eucl X Y)
  |  |-  context[ Z.div_eucl ?X ?Y ] =>
     generalize (Z_div_mod_full X Y) ; case (Z.div_eucl X Y)
  end; unfold Remainder.

Lemma sqrt_main_trick j k: 0 <= j -> 0 <= k -> (j * k) + j <= ((j + k)/2 + 1) ^ 2.