Module Import Private_NZDiv := Nop <+ NZDivProp A A B.
(** Another formulation of the main equation *)
(** We have a general bound for absolute values *)
(** Uniqueness theorems *)
Fact opp_mod_bound_or : forall a b, b~=0 ->
0 <= -(a mod b) < -b \/ -b < -(a mod b) <= 0.
(** With the current conventions, the other sign rules are rather complex. *)
(** The sign of [a mod b] is the one of [b] (when it isn't null) *)
(** A division by itself returns 1 *)
(** A division of a small number by a bigger one yields zero. *)
(** * Order results about mod and div *)
(** A modulo cannot grow beyond its starting point. *)
(** As soon as the divisor is strictly greater than 1, the division is strictly decreasing. *)
(** In this convention, [div] performs Rounding-Toward-Bottom.  Since we cannot speak of rational values here, we express this fact by multiplying back by [b], and this leads to separates statements according to the sign of [b].  First, [a/b] is below the exact fraction ... *)
(** ... and moreover it is the larger such integer, since [S(a/b)] is strictly above the exact fraction. *)
(** NB: The four previous properties could be used as specifications for [div]. *)
(** Inequality [mul_div_le] is exact iff the modulo is zero. *)
(** Some additional inequalities about div. *)
(** A division respects opposite monotonicity for the divisor *)
(** Cancellations. *)
(** Operations modulo. *)
(** With the current convention, the following result isn't always true with a negative last divisor. For instance [ 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) ], or [ 5/2/(-2) = -1 <> -2 = 5 / (2*-2) ]. *)
(** Similarly, the following result doesn't always hold when [c<0]. For instance [3 mod (-2*-2)) = 3] while [3 mod (-2) + (-2)*((3/-2) mod -2) = -1]. *)
(** A last inequality: *)

Theorem div_mul_le:
forall a b c, 0<=a -> 0<b -> 0<=c -> c*(a/b) <= (c*a)/b.