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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
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(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** %\subsection*{ extras :  before\_after.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export omit_facts.
(** - Some stuff I thought would be useful some day but wasn't *)
Definition before (N N' : nat) (i : fin N) (j : fin N') := index i < index j.
Definition after (N N' : nat) (i : fin N) (j : fin N') := index i > index j.
Definition notbefore (N N' : nat) (i : fin N) (j : fin N') :=
index i = index j \/ after i j.
Definition notafter (N N' : nat) (i : fin N) (j : fin N') :=
before i j \/ index i = index j.

Lemma decide_fin :
forall (N N' : nat) (i : fin N) (i' : fin N'),
before i i' \/ index i = index i' \/ after i i'.