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[helm.git] / matita / contribs / CoRN-Decl / model / setoids / Zsetoid.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/model/setoids/Zsetoid".
+
+include "CoRN.ma".
+
+(* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
+
+include "model/structures/Zsec.ma".
+
+include "algebra/CSetoidFun.ma".
+
+(*#* **Example of a setoid: [Z]
+*** [Z]
+*)
+
+inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con".
+
+(*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
+***Addition
+We will prove now that the addition on the integers is a setoid function.
+*)
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con".
+
+(*#* What's more: the addition is also associative and commutative.
+*)
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con".
+
+(*#* ***Opposite
+Taking the opposite of an integer is a setoid function.
+*)
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con".
+
+(*#* ***Multiplication
+Finally the multiplication is a setoid function and is associative and commutative.
+*)
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con".
+
+inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con".
+