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(* 02110-1301 USA                                                     *)
Unset Standard Proposition Elimination Names.
(** Divisibility *)
Inductive divide (a b : Z) : Prop :=
divide_intro : forall q : Z, b = (q * a)%Z -> divide a b.
Notation "( x | y )" := (divide x y) (at level 0) : Z_scope.
Local Open Scope Z_scope.
(** Results *)
Hint Resolve divide_refl one_divide divide_0.
Hint Resolve divide_mult_left divide_mult_right.
Hint Resolve divide_plus divide_opp divide_opp_rev divide_opp_left
divide_opp_left_rev divide_minus divide_left divide_right divide_a_ab
divide_a_ba.
(** Trivial lemmas to do case analysis over [x:Z]. *)
(** If [a] divides [b] and [b] divides [a] then [a] is [b] or [-b]. *)
(** If [a] divides [b] and [b<>0] then [|a| <= |b|]. *)

Lemma Zabs_ind :
forall (P : Z -> Prop) (x : Z),
(x >= 0 -> P x) -> (x <= 0 -> P (- x)) -> P (Z.abs x).