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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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/algebra/Expon".
21 (* $Id: Expon.v,v 1.5 2004/04/23 10:00:54 lcf Exp $ *)
23 (*#* printing [^^] %\ensuremath{\hat{\ }}% #^# *)
25 include "algebra/COrdCauchy.ma".
27 include "tactics/Transparent_algebra.ma".
30 **More properties about [nexp]
31 %\begin{convention}% Let [R] be an ordered field.
39 alias id "R" = "cic:/CoRN/algebra/Expon/More_Nexp/R.var".
41 inline "cic:/CoRN/algebra/Expon/nexp_resp_ap_zero.con".
44 Hint Resolve nexp_resp_ap_zero: algebra.
47 inline "cic:/CoRN/algebra/Expon/nexp_distr_div.con".
49 inline "cic:/CoRN/algebra/Expon/nexp_distr_div'.con".
51 inline "cic:/CoRN/algebra/Expon/small_nexp_resp_lt.con".
53 inline "cic:/CoRN/algebra/Expon/great_nexp_resp_lt.con".
55 inline "cic:/CoRN/algebra/Expon/small_nexp_resp_le.con".
57 inline "cic:/CoRN/algebra/Expon/great_nexp_resp_le.con".
59 inline "cic:/CoRN/algebra/Expon/nexp_resp_leEq.con".
61 inline "cic:/CoRN/algebra/Expon/nexp_resp_leEq_one.con".
63 inline "cic:/CoRN/algebra/Expon/nexp_resp_leEq_neg_even.con".
65 inline "cic:/CoRN/algebra/Expon/nexp_resp_leEq_neg_odd.con".
67 inline "cic:/CoRN/algebra/Expon/nexp_distr_recip.con".
70 Hint Resolve nexp_distr_recip: algebra.
73 inline "cic:/CoRN/algebra/Expon/nexp_even_nonneg.con".
75 inline "cic:/CoRN/algebra/Expon/nexp_resp_le'.con".
77 inline "cic:/CoRN/algebra/Expon/nexp_resp_le.con".
79 inline "cic:/CoRN/algebra/Expon/bin_less_un.con".
86 Hint Resolve nexp_distr_div nexp_distr_recip: algebra.
90 Implicit Arguments nexp_resp_ap_zero [R x].
93 (*#* **Definition of [zexp]: integer exponentiation
94 %\begin{convention}% Let [R] be an ordered field.
102 alias id "R" = "cic:/CoRN/algebra/Expon/Zexp_def/R.var".
105 It would be nicer to define [zexp] using [caseZdiff], but we already
106 have most properties now.
109 inline "cic:/CoRN/algebra/Expon/zexp.con".
116 Implicit Arguments zexp [R].
120 Notation "( x [//] Hx ) [^^] n" := (zexp x Hx n) (at level 0).
123 (*#* **Properties of [zexp]
124 %\begin{convention}% Let [R] be an ordered field.
129 Section Zexp_properties
132 alias id "R" = "cic:/CoRN/algebra/Expon/Zexp_properties/R.var".
134 inline "cic:/CoRN/algebra/Expon/zexp_zero.con".
137 Hint Resolve zexp_zero: algebra.
140 inline "cic:/CoRN/algebra/Expon/zexp_nexp.con".
143 Hint Resolve zexp_nexp: algebra.
146 inline "cic:/CoRN/algebra/Expon/zexp_inv_nexp.con".
149 Hint Resolve zexp_inv_nexp: algebra.
152 inline "cic:/CoRN/algebra/Expon/zexp_inv_nexp'.con".
155 Hint Resolve zexp_inv_nexp': algebra.
158 inline "cic:/CoRN/algebra/Expon/zexp_strext.con".
160 inline "cic:/CoRN/algebra/Expon/zexp_wd.con".
163 Hint Resolve zexp_wd: algebra_c.
166 inline "cic:/CoRN/algebra/Expon/zexp_plus1.con".
169 Hint Resolve zexp_plus1: algebra.
172 inline "cic:/CoRN/algebra/Expon/zexp_resp_ap_zero.con".
175 Hint Resolve zexp_resp_ap_zero: algebra.
178 inline "cic:/CoRN/algebra/Expon/zexp_inv.con".
181 Hint Resolve zexp_inv: algebra.
184 inline "cic:/CoRN/algebra/Expon/zexp_inv1.con".
187 Hint Resolve zexp_inv1: algebra.
190 inline "cic:/CoRN/algebra/Expon/zexp_plus.con".
193 Hint Resolve zexp_plus: algebra.
196 inline "cic:/CoRN/algebra/Expon/zexp_minus.con".
199 Hint Resolve zexp_minus: algebra.
202 inline "cic:/CoRN/algebra/Expon/one_zexp.con".
205 Hint Resolve one_zexp: algebra.
208 inline "cic:/CoRN/algebra/Expon/mult_zexp.con".
211 Hint Resolve mult_zexp: algebra.
214 inline "cic:/CoRN/algebra/Expon/zexp_mult.con".
217 Hint Resolve zexp_mult: algebra.
220 inline "cic:/CoRN/algebra/Expon/zexp_two.con".
223 Hint Resolve zexp_two: algebra.
226 inline "cic:/CoRN/algebra/Expon/inv_zexp_even.con".
229 Hint Resolve inv_zexp_even: algebra.
232 inline "cic:/CoRN/algebra/Expon/inv_zexp_two.con".
235 Hint Resolve inv_zexp_two: algebra.
238 inline "cic:/CoRN/algebra/Expon/inv_zexp_odd.con".
240 inline "cic:/CoRN/algebra/Expon/zexp_one.con".
243 Hint Resolve zexp_one: algebra.
246 inline "cic:/CoRN/algebra/Expon/zexp_funny.con".
249 Hint Resolve zexp_funny: algebra.
252 inline "cic:/CoRN/algebra/Expon/zexp_funny'.con".
255 Hint Resolve zexp_funny': algebra.
258 inline "cic:/CoRN/algebra/Expon/zexp_pos.con".
265 Hint Resolve nexp_resp_ap_zero zexp_zero zexp_nexp zexp_inv_nexp
266 zexp_inv_nexp' zexp_plus1 zexp_resp_ap_zero zexp_inv zexp_inv1 zexp_plus
267 zexp_minus one_zexp mult_zexp zexp_mult zexp_two inv_zexp_even inv_zexp_two
268 zexp_one zexp_funny zexp_funny': algebra.
272 Hint Resolve zexp_wd: algebra_c.
279 alias id "R" = "cic:/CoRN/algebra/Expon/Root_Unique/R.var".
281 inline "cic:/CoRN/algebra/Expon/root_unique.con".
283 inline "cic:/CoRN/algebra/Expon/root_one.con".