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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 (* $Id: CGroups.v,v 1.9 2004/04/23 10:00:52 lcf Exp $ *)
21 (*#* printing [-] %\ensuremath-% #−# *)
23 (*#* printing [--] %\ensuremath-% #−# *)
25 (*#* printing {-} %\ensuremath-% #−# *)
27 (*#* printing {--} %\ensuremath-% #−# *)
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.
55 Implicit Arguments cg_inv [c].
58 inline cic:/CoRN/algebra/CGroups/cg_minus.con.
61 %\begin{nameconvention}%
62 In the names of lemmas, we will denote [[--] ] with [inv],
63 and [ [-] ] with [minus].
64 %\end{nameconvention}%
68 Implicit Arguments cg_minus [G].
75 %\begin{convention}% Let [G] be a group.
80 Section CGroup_axioms.
83 inline cic:/CoRN/algebra/CGroups/G.var.
85 inline cic:/CoRN/algebra/CGroups/cg_inverse.con.
93 General properties of groups.
94 %\begin{convention}% Let [G] be a group.
99 Section CGroup_basics.
102 inline cic:/CoRN/algebra/CGroups/G.var.
104 inline cic:/CoRN/algebra/CGroups/cg_rht_inv_unfolded.con.
106 inline cic:/CoRN/algebra/CGroups/cg_lft_inv_unfolded.con.
108 inline cic:/CoRN/algebra/CGroups/cg_minus_correct.con.
111 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded cg_minus_correct:
115 inline cic:/CoRN/algebra/CGroups/cg_inverse'.con.
119 inline cic:/CoRN/algebra/CGroups/cg_minus_unfolded.con.
122 Hint Resolve cg_minus_unfolded: algebra.
125 inline cic:/CoRN/algebra/CGroups/cg_minus_wd.con.
128 Hint Resolve cg_minus_wd: algebra_c.
131 inline cic:/CoRN/algebra/CGroups/cg_minus_strext.con.
133 inline cic:/CoRN/algebra/CGroups/cg_minus_is_csetoid_bin_op.con.
135 inline cic:/CoRN/algebra/CGroups/grp_inv_assoc.con.
138 Hint Resolve grp_inv_assoc: algebra.
141 inline cic:/CoRN/algebra/CGroups/cg_inv_unique.con.
143 inline cic:/CoRN/algebra/CGroups/cg_inv_inv.con.
146 Hint Resolve cg_inv_inv: algebra.
149 inline cic:/CoRN/algebra/CGroups/cg_cancel_lft.con.
151 inline cic:/CoRN/algebra/CGroups/cg_cancel_rht.con.
153 inline cic:/CoRN/algebra/CGroups/cg_inv_unique'.con.
155 inline cic:/CoRN/algebra/CGroups/cg_inv_unique_2.con.
157 inline cic:/CoRN/algebra/CGroups/cg_zero_inv.con.
160 Hint Resolve cg_zero_inv: algebra.
163 inline cic:/CoRN/algebra/CGroups/cg_inv_zero.con.
165 inline cic:/CoRN/algebra/CGroups/cg_inv_op.con.
168 Useful for interactive proof development.
171 inline cic:/CoRN/algebra/CGroups/x_minus_x.con.
175 %\begin{convention}% Let [P] be a predicate on [G] containing
176 [Zero] and closed under [[+]] and [[--] ].
184 inline cic:/CoRN/algebra/CGroups/P.var.
186 inline cic:/CoRN/algebra/CGroups/Punit.var.
188 inline cic:/CoRN/algebra/CGroups/op_pres_P.var.
190 inline cic:/CoRN/algebra/CGroups/inv_pres_P.var.
192 inline cic:/CoRN/algebra/CGroups/subcrr.con.
194 inline cic:/CoRN/algebra/CGroups/subinv.con.
196 inline cic:/CoRN/algebra/CGroups/isgrp_scrr.con.
198 inline cic:/CoRN/algebra/CGroups/Build_SubCGroup.con.
209 Hint Resolve cg_rht_inv_unfolded cg_lft_inv_unfolded: algebra.
213 Hint Resolve cg_inv_inv cg_minus_correct cg_zero_inv cg_inv_zero: algebra.
217 Hint Resolve cg_minus_unfolded grp_inv_assoc cg_inv_op: algebra.
221 Hint Resolve cg_minus_wd: algebra_c.
225 ** Associative properties of groups
226 %\begin{convention}% Let [G] be a group.
231 Section Assoc_properties.
234 inline cic:/CoRN/algebra/CGroups/G.var.
236 inline cic:/CoRN/algebra/CGroups/assoc_2.con.
238 inline cic:/CoRN/algebra/CGroups/zero_minus.con.
240 inline cic:/CoRN/algebra/CGroups/cg_cancel_mixed.con.
242 inline cic:/CoRN/algebra/CGroups/plus_resp_eq.con.
245 End Assoc_properties.
249 Hint Resolve assoc_2 minus_plus zero_minus cg_cancel_mixed plus_resp_eq:
254 ** Apartness in Constructive Groups
255 Specific properties of apartness.
256 %\begin{convention}% Let [G] be a group.
261 Section cgroups_apartness.
264 inline cic:/CoRN/algebra/CGroups/G.var.
266 inline cic:/CoRN/algebra/CGroups/cg_add_ap_zero.con.
268 inline cic:/CoRN/algebra/CGroups/op_rht_resp_ap.con.
270 inline cic:/CoRN/algebra/CGroups/cg_ap_cancel_rht.con.
272 inline cic:/CoRN/algebra/CGroups/plus_cancel_ap_rht.con.
274 inline cic:/CoRN/algebra/CGroups/minus_ap_zero.con.
276 inline cic:/CoRN/algebra/CGroups/zero_minus_apart.con.
278 inline cic:/CoRN/algebra/CGroups/inv_resp_ap_zero.con.
280 inline cic:/CoRN/algebra/CGroups/inv_resp_ap.con.
282 inline cic:/CoRN/algebra/CGroups/minus_resp_ap_rht.con.
284 inline cic:/CoRN/algebra/CGroups/minus_resp_ap_lft.con.
286 inline cic:/CoRN/algebra/CGroups/minus_cancel_ap_rht.con.
289 End cgroups_apartness.
293 Hint Resolve op_rht_resp_ap: algebra.
297 Hint Resolve minus_ap_zero zero_minus_apart inv_resp_ap_zero: algebra.
305 ** Functional operations
307 As before, we lift our group operations to the function space of the group.
310 Let [G] be a group and [F,F':(PartFunct G)] with domains given respectively
315 inline cic:/CoRN/algebra/CGroups/G.var.
317 inline cic:/CoRN/algebra/CGroups/F.var.
319 inline cic:/CoRN/algebra/CGroups/F'.var.
323 inline cic:/CoRN/algebra/CGroups/P.con.
325 inline cic:/CoRN/algebra/CGroups/Q.con.
330 Section Part_Function_Inv.
333 inline cic:/CoRN/algebra/CGroups/part_function_inv_strext.con.
335 inline cic:/CoRN/algebra/CGroups/Finv.con.
338 End Part_Function_Inv.
342 Section Part_Function_Minus.
345 inline cic:/CoRN/algebra/CGroups/part_function_minus_strext.con.
347 inline cic:/CoRN/algebra/CGroups/Fminus.con.
350 End Part_Function_Minus.
354 %\begin{convention}% Let [R:G->CProp].
358 inline cic:/CoRN/algebra/CGroups/R.var.
360 inline cic:/CoRN/algebra/CGroups/included_FInv.con.
362 inline cic:/CoRN/algebra/CGroups/included_FInv'.con.
364 inline cic:/CoRN/algebra/CGroups/included_FMinus.con.
366 inline cic:/CoRN/algebra/CGroups/included_FMinus'.con.
368 inline cic:/CoRN/algebra/CGroups/included_FMinus''.con.
375 Implicit Arguments Finv [G].
379 Implicit Arguments Fminus [G].
383 Hint Resolve included_FInv included_FMinus : included.
387 Hint Immediate included_FInv' included_FMinus' included_FMinus'' : included.