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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/algebra/CSemiGroups".
21 (* $Id: CSemiGroups.v,v 1.8 2004/04/22 14:49:43 lcf Exp $ *)
23 (*#* printing [+] %\ensuremath+% #+# *)
25 (*#* printing {+} %\ensuremath+% #+# *)
27 include "algebra/CSetoidInc.ma".
33 **Definition of the notion of semigroup
36 inline "cic:/CoRN/algebra/CSemiGroups/is_CSemiGroup.con".
38 inline "cic:/CoRN/algebra/CSemiGroups/CSemiGroup.ind".
40 coercion "cic:/matita/CoRN-Decl/algebra/CSemiGroups/csg_crr.con" 0 (* compounds *).
43 %\begin{nameconvention}%
44 In the %{\em %names%}% of lemmas, we will denote [[+]] with [plus].
45 %\end{nameconvention}%
49 Implicit Arguments csg_op [c].
54 (*#* **Semigroup axioms
55 The axiomatic properties of a semi group.
57 %\begin{convention}% Let [G] be a semi-group.
62 Section CSemiGroup_axioms.
65 inline "cic:/CoRN/algebra/CSemiGroups/G.var".
67 inline "cic:/CoRN/algebra/CSemiGroups/CSemiGroup_is_CSemiGroup.con".
69 inline "cic:/CoRN/algebra/CSemiGroups/plus_assoc.con".
72 End CSemiGroup_axioms.
77 (*#* **Semigroup basics
80 Let [G] be a semi-group.
85 Section CSemiGroup_basics.
88 inline "cic:/CoRN/algebra/CSemiGroups/G.var".
92 inline "cic:/CoRN/algebra/CSemiGroups/plus_assoc_unfolded.con".
95 End CSemiGroup_basics.
101 Hint Resolve plus_assoc_unfolded: algebra.
104 (*#* **Functional operations
105 We can also define a similar addition operator, which will be denoted by [{+}], on partial functions.
107 %\begin{convention}% Whenever possible, we will denote the functional construction corresponding to an algebraic operation by the same symbol enclosed in curly braces.
110 At this stage, we will always consider automorphisms; we %{\em %could%}% treat this in a more general setting, but we feel that it wouldn't really be a useful effort.
112 %\begin{convention}% Let [G:CSemiGroup] and [F,F':(PartFunct G)] and denote by [P] and [Q], respectively, the predicates characterizing their domains.
117 Section Part_Function_Plus.
120 inline "cic:/CoRN/algebra/CSemiGroups/G.var".
122 inline "cic:/CoRN/algebra/CSemiGroups/F.var".
124 inline "cic:/CoRN/algebra/CSemiGroups/F'.var".
128 inline "cic:/CoRN/algebra/CSemiGroups/P.con".
130 inline "cic:/CoRN/algebra/CSemiGroups/Q.con".
134 inline "cic:/CoRN/algebra/CSemiGroups/part_function_plus_strext.con".
136 inline "cic:/CoRN/algebra/CSemiGroups/Fplus.con".
139 %\begin{convention}% Let [R:G->CProp].
143 inline "cic:/CoRN/algebra/CSemiGroups/R.var".
145 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus.con".
147 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus'.con".
149 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus''.con".
152 End Part_Function_Plus.
156 Implicit Arguments Fplus [G].
160 Hint Resolve included_FPlus : included.
164 Hint Immediate included_FPlus' included_FPlus'' : included.
169 Let [csg] a semi-group and [P] a non-empty
170 predicate on the semi-group which is preserved by [[+]].
175 Section SubCSemiGroups.
178 inline "cic:/CoRN/algebra/CSemiGroups/csg.var".
180 inline "cic:/CoRN/algebra/CSemiGroups/P.var".
182 inline "cic:/CoRN/algebra/CSemiGroups/op_pres_P.var".
184 inline "cic:/CoRN/algebra/CSemiGroups/subcrr.con".
186 inline "cic:/CoRN/algebra/CSemiGroups/Build_SubCSemiGroup.con".