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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/model/setoids/Zsetoid".
19 include "CoRN_notation.ma".
21 (* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
23 include "model/structures/Zsec.ma".
25 include "algebra/CSetoidFun.ma".
27 (*#* **Example of a setoid: [Z]
31 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con".
33 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con".
35 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con".
37 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con".
39 inline "cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con".
41 inline "cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con".
43 (*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
45 We will prove now that the addition on the integers is a setoid function.
48 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con".
50 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con".
52 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con".
54 (*#* What's more: the addition is also associative and commutative.
57 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con".
59 inline "cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con".
62 Taking the opposite of an integer is a setoid function.
65 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con".
67 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con".
69 inline "cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con".
71 (*#* ***Multiplication
72 Finally the multiplication is a setoid function and is associative and commutative.
75 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con".
77 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con".
79 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con".
81 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con".
83 inline "cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con".