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[helm.git] / matita / contribs / CoRN-Decl / model / setoids / Qpossetoid.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/Qpossetoid".
+
+include "CoRN.ma".
+
+(* $Id: Qpossetoid.v,v 1.4 2004/04/06 15:46:05 lcf Exp $ *)
+
+include "model/setoids/Qsetoid.ma".
+
+include "algebra/CSetoidFun.ma".
+
+include "model/structures/Qpossec.ma".
+
+(*#* **Example of a setoid: [Qpos]
+***Setoid
+We will examine the subsetoid of positive rationals of the setoid of 
+rational numbers.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/QposP.con".
+
+(*#* One, two and four are elements of it.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/QONEpos.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/QTWOpos.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/QFOURpos.con".
+
+(*#* ***Multiplication
+As we have seen, multiplication preserves positivity, so we can restrict it
+ to the positive rationals. We see that this restricted multiplication has some
+ nice properties.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qmult_pres_pos1.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_mult.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/associative_Qpos_mult.con".
+
+(*#* ***Inverse
+We restrict the domain of the inverse to the set of positive rationals.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_inv.con".
+
+(*#* The restricted inverse preserves positivity.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/inv_pres_pos1.con".
+
+(*#* Now, we can also restrict the co-domain.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv.con".
+
+(*#* This restricted inverse map appears a setoid function.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv_strong_ext.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_Qpos_inv_op.con".
+
+(*#* ***Special multiplication and inverse
+We define [multdiv2]: $(x,y) \mapsto xy/2$ #(x,y) ↦ xy/2#.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/Qpos_div2.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/multdiv2.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/associative_multdiv2.con".
+
+(*#* And its inverse [multdiv4]: $x \mapsto 4/x$ #x ↦ 4/x#.
+*)
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/mult4.con".
+
+inline "cic:/CoRN/model/setoids/Qpossetoid/divmult4.con".
+