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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 (* $Id: Zsetoid.v,v 1.5 2004/04/07 15:08:08 lcf Exp $ *)
29 (*#* **Example of a setoid: [Z]
33 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_irreflexive.con.
35 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_symmetric.con.
37 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_cotransitive.con.
39 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_tight.con.
41 inline cic:/CoRN/model/setoids/Zsetoid/ap_Z_is_apartness.con.
43 inline cic:/CoRN/model/setoids/Zsetoid/Z_as_CSetoid.con.
45 (*#* The term [Z_as_CSetoid] is of type [CSetoid]. Hence we have proven that [Z] is a constructive setoid.
47 We will prove now that the addition on the integers is a setoid function.
50 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_wd.con.
52 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_strext.con.
54 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_bin_fun.con.
56 (*#* What's more: the addition is also associative and commutative.
59 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_assoc.con.
61 inline cic:/CoRN/model/setoids/Zsetoid/Zplus_is_commut.con.
64 Taking the opposite of an integer is a setoid function.
67 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_wd.con.
69 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_strext.con.
71 inline cic:/CoRN/model/setoids/Zsetoid/Zopp_is_fun.con.
73 (*#* ***Multiplication
74 Finally the multiplication is a setoid function and is associative and commutative.
77 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_wd.con.
79 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_strext.con.
81 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_bin_fun.con.
83 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_assoc.con.
85 inline cic:/CoRN/model/setoids/Zsetoid/Zmult_is_commut.con.