helm/software/matita/contribs/procedural/CoRN/algebra/CSemiGroups.mma

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   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 (* This file was automatically generated: do not edit *********************)
  16
  17 include "CoRN.ma".
  18
  19 (* $Id: CSemiGroups.v,v 1.8 2004/04/22 14:49:43 lcf Exp $ *)
  20
  21 (*#* printing [+] %\ensuremath+% #+# *)
  22
  23 (*#* printing {+} %\ensuremath+% #+# *)
  24
  25 include "algebra/CSetoidInc.ma".
  26
  27 (* Begin_SpecReals *)
  28
  29 (*#* *Semigroups
  30
  31 **Definition of the notion of semigroup
  32 *)
  33
  34 inline procedural "cic:/CoRN/algebra/CSemiGroups/is_CSemiGroup.con" as definition.
  35
  36 inline procedural "cic:/CoRN/algebra/CSemiGroups/CSemiGroup.ind".
  37
  38 (* COERCION
  39 cic:/matita/CoRN-Procedural/algebra/CSemiGroups/csg_crr.con
  40 *)
  41
  42 (*#*
  43 %\begin{nameconvention}%
  44 In the %{\em %names%}% of lemmas, we will denote [[+]] with [plus].
  45 %\end{nameconvention}%
  46 *)
  47
  48 (* UNEXPORTED
  49 Implicit Arguments csg_op [c].
  50 *)
  51
  52 (* NOTATION
  53 Infix "[+]" := csg_op (at level 50, left associativity).
  54 *)
  55
  56 (* End_SpecReals *)
  57
  58 (*#* **Semigroup axioms
  59 The axiomatic properties of a semi group.
  60
  61 %\begin{convention}% Let [G] be a semi-group.
  62 %\end{convention}%
  63 *)
  64
  65 (* UNEXPORTED
  66 Section CSemiGroup_axioms
  67 *)
  68
  69 (* UNEXPORTED
  70 cic:/CoRN/algebra/CSemiGroups/CSemiGroup_axioms/G.var
  71 *)
  72
  73 inline procedural "cic:/CoRN/algebra/CSemiGroups/CSemiGroup_is_CSemiGroup.con" as lemma.
  74
  75 inline procedural "cic:/CoRN/algebra/CSemiGroups/plus_assoc.con" as lemma.
  76
  77 (* UNEXPORTED
  78 End CSemiGroup_axioms
  79 *)
  80
  81 (* Begin_SpecReals *)
  82
  83 (*#* **Semigroup basics
  84
  85 %\begin{convention}%
  86 Let [G] be a semi-group.
  87 %\end{convention}%
  88 *)
  89
  90 (* UNEXPORTED
  91 Section CSemiGroup_basics
  92 *)
  93
  94 (* UNEXPORTED
  95 cic:/CoRN/algebra/CSemiGroups/CSemiGroup_basics/G.var
  96 *)
  97
  98 (* End_SpecReals *)
  99
 100 inline procedural "cic:/CoRN/algebra/CSemiGroups/plus_assoc_unfolded.con" as lemma.
 101
 102 (* UNEXPORTED
 103 End CSemiGroup_basics
 104 *)
 105
 106 (* End_SpecReals *)
 107
 108 (* UNEXPORTED
 109 Hint Resolve plus_assoc_unfolded: algebra.
 110 *)
 111
 112 (*#* **Functional operations
 113 We can also define a similar addition operator, which will be denoted by [{+}], on partial functions.
 114
 115 %\begin{convention}% Whenever possible, we will denote the functional construction corresponding to an algebraic operation by the same symbol enclosed in curly braces.
 116 %\end{convention}%
 117
 118 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.
 119
 120 %\begin{convention}% Let [G:CSemiGroup] and [F,F':(PartFunct G)] and denote by [P] and [Q], respectively, the predicates characterizing their domains.
 121 %\end{convention}%
 122 *)
 123
 124 (* UNEXPORTED
 125 Section Part_Function_Plus
 126 *)
 127
 128 (* UNEXPORTED
 129 cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/G.var
 130 *)
 131
 132 (* UNEXPORTED
 133 cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/F.var
 134 *)
 135
 136 (* UNEXPORTED
 137 cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/F'.var
 138 *)
 139
 140 (* begin hide *)
 141
 142 inline procedural "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/P.con" "Part_Function_Plus__" as definition.
 143
 144 inline procedural "cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/Q.con" "Part_Function_Plus__" as definition.
 145
 146 (* end hide *)
 147
 148 inline procedural "cic:/CoRN/algebra/CSemiGroups/part_function_plus_strext.con" as lemma.
 149
 150 inline procedural "cic:/CoRN/algebra/CSemiGroups/Fplus.con" as definition.
 151
 152 (*#*
 153 %\begin{convention}% Let [R:G->CProp].
 154 %\end{convention}%
 155 *)
 156
 157 (* UNEXPORTED
 158 cic:/CoRN/algebra/CSemiGroups/Part_Function_Plus/R.var
 159 *)
 160
 161 inline procedural "cic:/CoRN/algebra/CSemiGroups/included_FPlus.con" as lemma.
 162
 163 inline procedural "cic:/CoRN/algebra/CSemiGroups/included_FPlus'.con" as lemma.
 164
 165 inline procedural "cic:/CoRN/algebra/CSemiGroups/included_FPlus''.con" as lemma.
 166
 167 (* UNEXPORTED
 168 End Part_Function_Plus
 169 *)
 170
 171 (* UNEXPORTED
 172 Implicit Arguments Fplus [G].
 173 *)
 174
 175 (* NOTATION
 176 Infix "{+}" := Fplus (at level 50, left associativity).
 177 *)
 178
 179 (* UNEXPORTED
 180 Hint Resolve included_FPlus : included.
 181 *)
 182
 183 (* UNEXPORTED
 184 Hint Immediate included_FPlus' included_FPlus'' : included.
 185 *)
 186
 187 (*#* **Subsemigroups
 188 %\begin{convention}%
 189 Let [csg] a semi-group and [P] a non-empty
 190 predicate on the semi-group which is preserved by [[+]].
 191 %\end{convention}%
 192 *)
 193
 194 (* UNEXPORTED
 195 Section SubCSemiGroups
 196 *)
 197
 198 (* UNEXPORTED
 199 cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/csg.var
 200 *)
 201
 202 (* UNEXPORTED
 203 cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/P.var
 204 *)
 205
 206 (* UNEXPORTED
 207 cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/op_pres_P.var
 208 *)
 209
 210 inline procedural "cic:/CoRN/algebra/CSemiGroups/SubCSemiGroups/subcrr.con" "SubCSemiGroups__" as definition.
 211
 212 inline procedural "cic:/CoRN/algebra/CSemiGroups/Build_SubCSemiGroup.con" as definition.
 213
 214 (* UNEXPORTED
 215 End SubCSemiGroups
 216 *)
 217