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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-
-(*i $Id: Compare.v,v 1.1.2.1 2004/07/16 19:31:23 herbelin Exp $ i*)
-
-(** Equality is decidable on [nat] *)
-V7only [Import nat_scope.].
-Open Local Scope nat_scope.
-
-(*
-Lemma not_eq_sym : (A:Set)(p,q:A)(~p=q) -> ~(q=p).
-Proof sym_not_eq.
-Hints Immediate not_eq_sym : arith.
-*)
-Notation not_eq_sym := sym_not_eq.
-
-Implicit Variables Type m,n,p,q:nat.
-
-Require Arith.
-Require Peano_dec.
-Require Compare_dec.
-
-Definition le_or_le_S := le_le_S_dec.
-
-Definition compare := gt_eq_gt_dec.
-
-Lemma le_dec : (n,m:nat) {le n m} + {le m n}.
-Proof le_ge_dec.
-
-Definition lt_or_eq := [n,m:nat]{(gt m n)}+{n=m}.
-
-Lemma le_decide : (n,m:nat)(le n m)->(lt_or_eq n m).
-Proof le_lt_eq_dec.
-
-Lemma le_le_S_eq : (p,q:nat)(le p q)->((le (S p) q)\/(p=q)).
-Proof le_lt_or_eq.
-
-(* By special request of G. Kahn - Used in Group Theory *)
-Lemma discrete_nat : (m, n: nat) (lt m n) ->
-   (S m) = n \/ (EX r: nat | n = (S (S (plus m r)))).
-Proof.
-Intros m n H.
-LApply (lt_le_S m n); Auto with arith.
-Intro H'; LApply (le_lt_or_eq (S m) n); Auto with arith.
-NewInduction 1; Auto with arith.
-Right; Exists (minus n (S (S m))); Simpl.
-Rewrite (plus_sym m (minus n (S (S m)))).
-Rewrite (plus_n_Sm (minus n (S (S m))) m).
-Rewrite (plus_n_Sm (minus n (S (S m))) (S m)).
-Rewrite (plus_sym (minus n (S (S m))) (S (S m))); Auto with arith.
-Qed.
-
-Require Export Wf_nat.
-
-Require Export Min.