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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].
53 Infix "[+]" := csg_op (at level 50, left associativity).
58 (*#* **Semigroup axioms
59 The axiomatic properties of a semi group.
61 %\begin{convention}% Let [G] be a semi-group.
66 Section CSemiGroup_axioms
69 alias id "G" = "cic:/CoRN/algebra/CSemiGroups/CSemiGroup_axioms/G.var".
71 inline "cic:/CoRN/algebra/CSemiGroups/CSemiGroup_is_CSemiGroup.con".
73 inline "cic:/CoRN/algebra/CSemiGroups/plus_assoc.con".
81 (*#* **Semigroup basics
84 Let [G] be a semi-group.
89 Section CSemiGroup_basics
92 alias id "G" = "cic:/CoRN/algebra/CSemiGroups/CSemiGroup_basics/G.var".
96 inline "cic:/CoRN/algebra/CSemiGroups/plus_assoc_unfolded.con".
105 Hint Resolve plus_assoc_unfolded: algebra.
108 (*#* **Functional operations
109 We can also define a similar addition operator, which will be denoted by [{+}], on partial functions.
111 %\begin{convention}% Whenever possible, we will denote the functional construction corresponding to an algebraic operation by the same symbol enclosed in curly braces.
114 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.
116 %\begin{convention}% Let [G:CSemiGroup] and [F,F':(PartFunct G)] and denote by [P] and [Q], respectively, the predicates characterizing their domains.
121 Section Part_Function_Plus
124 alias id "G" = "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/G.var".
126 alias id "F" = "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/F.var".
128 alias id "F'" = "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/F'.var".
132 inline "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/P.con" "Part_Function_Plus__".
134 inline "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/Q.con" "Part_Function_Plus__".
138 inline "cic:/CoRN/algebra/CSemiGroups/part_function_plus_strext.con".
140 inline "cic:/CoRN/algebra/CSemiGroups/Fplus.con".
143 %\begin{convention}% Let [R:G->CProp].
147 alias id "R" = "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/R.var".
149 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus.con".
151 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus'.con".
153 inline "cic:/CoRN/algebra/CSemiGroups/included_FPlus''.con".
156 End Part_Function_Plus
160 Implicit Arguments Fplus [G].
164 Infix "{+}" := Fplus (at level 50, left associativity).
168 Hint Resolve included_FPlus : included.
172 Hint Immediate included_FPlus' included_FPlus'' : included.
177 Let [csg] a semi-group and [P] a non-empty
178 predicate on the semi-group which is preserved by [[+]].
183 Section SubCSemiGroups
186 alias id "csg" = "cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/csg.var".
188 alias id "P" = "cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/P.var".
190 alias id "op_pres_P" = "cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/op_pres_P.var".
192 inline "cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/subcrr.con" "SubCSemiGroups__".
194 inline "cic:/CoRN/algebra/CSemiGroups/Build_SubCSemiGroup.con".