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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/CGroups".
19 include "CoRN_notation.ma".
21 (* $Id: CGroups.v,v 1.9 2004/04/23 10:00:52 lcf Exp $ *)
23 (*#* printing [-] %\ensuremath-% #−# *)
25 (*#* printing [--] %\ensuremath-% #−# *)
27 (*#* printing {-} %\ensuremath-% #−# *)
29 (*#* printing {--} %\ensuremath-% #−# *)
31 include "algebra/CMonoids.ma".
37 ** Definition of the notion of Group
40 inline "cic:/CoRN/algebra/CGroups/is_inverse.con".
43 Implicit Arguments is_inverse [S].
46 inline "cic:/CoRN/algebra/CGroups/is_CGroup.con".
48 inline "cic:/CoRN/algebra/CGroups/CGroup.ind".
50 coercion "cic:/matita/CoRN-Decl/algebra/CGroups/cg_crr.con" 0 (* compounds *).
57 Implicit Arguments cg_inv [c].
60 inline "cic:/CoRN/algebra/CGroups/cg_minus.con".
63 %\begin{nameconvention}%
64 In the names of lemmas, we will denote [[--] ] with [inv],
65 and [ [-] ] with [minus].
66 %\end{nameconvention}%
70 Implicit Arguments cg_minus [G].
77 %\begin{convention}% Let [G] be a group.
82 Section CGroup_axioms.
85 inline "cic:/CoRN/algebra/CGroups/G.var".
87 inline "cic:/CoRN/algebra/CGroups/cg_inverse.con".
95 General properties of groups.
96 %\begin{convention}% Let [G] be a group.
101 Section CGroup_basics.
104 inline "cic:/CoRN/algebra/CGroups/G.var".
106 inline "cic:/CoRN/algebra/CGroups/cg_rht_inv_unfolded.con".
108 inline "cic:/CoRN/algebra/CGroups/cg_lft_inv_unfolded.con".
110 inline "cic:/CoRN/algebra/CGroups/cg_minus_correct.con".
113 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded cg_minus_correct:
117 inline "cic:/CoRN/algebra/CGroups/cg_inverse'.con".
121 inline "cic:/CoRN/algebra/CGroups/cg_minus_unfolded.con".
124 Hint Resolve cg_minus_unfolded: algebra.
127 inline "cic:/CoRN/algebra/CGroups/cg_minus_wd.con".
130 Hint Resolve cg_minus_wd: algebra_c.
133 inline "cic:/CoRN/algebra/CGroups/cg_minus_strext.con".
135 inline "cic:/CoRN/algebra/CGroups/cg_minus_is_csetoid_bin_op.con".
137 inline "cic:/CoRN/algebra/CGroups/grp_inv_assoc.con".
140 Hint Resolve grp_inv_assoc: algebra.
143 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique.con".
145 inline "cic:/CoRN/algebra/CGroups/cg_inv_inv.con".
148 Hint Resolve cg_inv_inv: algebra.
151 inline "cic:/CoRN/algebra/CGroups/cg_cancel_lft.con".
153 inline "cic:/CoRN/algebra/CGroups/cg_cancel_rht.con".
155 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique'.con".
157 inline "cic:/CoRN/algebra/CGroups/cg_inv_unique_2.con".
159 inline "cic:/CoRN/algebra/CGroups/cg_zero_inv.con".
162 Hint Resolve cg_zero_inv: algebra.
165 inline "cic:/CoRN/algebra/CGroups/cg_inv_zero.con".
167 inline "cic:/CoRN/algebra/CGroups/cg_inv_op.con".
170 Useful for interactive proof development.
173 inline "cic:/CoRN/algebra/CGroups/x_minus_x.con".
177 %\begin{convention}% Let [P] be a predicate on [G] containing
178 [Zero] and closed under [[+]] and [[--] ].
186 inline "cic:/CoRN/algebra/CGroups/P.var".
188 inline "cic:/CoRN/algebra/CGroups/Punit.var".
190 inline "cic:/CoRN/algebra/CGroups/op_pres_P.var".
192 inline "cic:/CoRN/algebra/CGroups/inv_pres_P.var".
194 inline "cic:/CoRN/algebra/CGroups/subcrr.con".
196 inline "cic:/CoRN/algebra/CGroups/subinv.con".
198 inline "cic:/CoRN/algebra/CGroups/isgrp_scrr.con".
200 inline "cic:/CoRN/algebra/CGroups/Build_SubCGroup.con".
211 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded: algebra.
215 Hint Resolve cg_inv_inv cg_minus_correct cg_zero_inv cg_inv_zero: algebra.
219 Hint Resolve cg_minus_unfolded grp_inv_assoc cg_inv_op: algebra.
223 Hint Resolve cg_minus_wd: algebra_c.
227 ** Associative properties of groups
228 %\begin{convention}% Let [G] be a group.
233 Section Assoc_properties.
236 inline "cic:/CoRN/algebra/CGroups/G.var".
238 inline "cic:/CoRN/algebra/CGroups/assoc_2.con".
240 inline "cic:/CoRN/algebra/CGroups/zero_minus.con".
242 inline "cic:/CoRN/algebra/CGroups/cg_cancel_mixed.con".
244 inline "cic:/CoRN/algebra/CGroups/plus_resp_eq.con".
247 End Assoc_properties.
251 Hint Resolve assoc_2 minus_plus zero_minus cg_cancel_mixed plus_resp_eq:
256 ** Apartness in Constructive Groups
257 Specific properties of apartness.
258 %\begin{convention}% Let [G] be a group.
263 Section cgroups_apartness.
266 inline "cic:/CoRN/algebra/CGroups/G.var".
268 inline "cic:/CoRN/algebra/CGroups/cg_add_ap_zero.con".
270 inline "cic:/CoRN/algebra/CGroups/op_rht_resp_ap.con".
272 inline "cic:/CoRN/algebra/CGroups/cg_ap_cancel_rht.con".
274 inline "cic:/CoRN/algebra/CGroups/plus_cancel_ap_rht.con".
276 inline "cic:/CoRN/algebra/CGroups/minus_ap_zero.con".
278 inline "cic:/CoRN/algebra/CGroups/zero_minus_apart.con".
280 inline "cic:/CoRN/algebra/CGroups/inv_resp_ap_zero.con".
282 inline "cic:/CoRN/algebra/CGroups/inv_resp_ap.con".
284 inline "cic:/CoRN/algebra/CGroups/minus_resp_ap_rht.con".
286 inline "cic:/CoRN/algebra/CGroups/minus_resp_ap_lft.con".
288 inline "cic:/CoRN/algebra/CGroups/minus_cancel_ap_rht.con".
291 End cgroups_apartness.
295 Hint Resolve op_rht_resp_ap: algebra.
299 Hint Resolve minus_ap_zero zero_minus_apart inv_resp_ap_zero: algebra.
307 ** Functional operations
309 As before, we lift our group operations to the function space of the group.
312 Let [G] be a group and [F,F':(PartFunct G)] with domains given respectively
317 inline "cic:/CoRN/algebra/CGroups/G.var".
319 inline "cic:/CoRN/algebra/CGroups/F.var".
321 inline "cic:/CoRN/algebra/CGroups/F'.var".
325 inline "cic:/CoRN/algebra/CGroups/P.con".
327 inline "cic:/CoRN/algebra/CGroups/Q.con".
332 Section Part_Function_Inv.
335 inline "cic:/CoRN/algebra/CGroups/part_function_inv_strext.con".
337 inline "cic:/CoRN/algebra/CGroups/Finv.con".
340 End Part_Function_Inv.
344 Section Part_Function_Minus.
347 inline "cic:/CoRN/algebra/CGroups/part_function_minus_strext.con".
349 inline "cic:/CoRN/algebra/CGroups/Fminus.con".
352 End Part_Function_Minus.
356 %\begin{convention}% Let [R:G->CProp].
360 inline "cic:/CoRN/algebra/CGroups/R.var".
362 inline "cic:/CoRN/algebra/CGroups/included_FInv.con".
364 inline "cic:/CoRN/algebra/CGroups/included_FInv'.con".
366 inline "cic:/CoRN/algebra/CGroups/included_FMinus.con".
368 inline "cic:/CoRN/algebra/CGroups/included_FMinus'.con".
370 inline "cic:/CoRN/algebra/CGroups/included_FMinus''.con".
377 Implicit Arguments Finv [G].
381 Implicit Arguments Fminus [G].
385 Hint Resolve included_FInv included_FMinus : included.
389 Hint Immediate included_FInv' included_FMinus' included_FMinus'' : included.