1 (**************************************************************************)
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".
19 include "CoRN_notation.ma".
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 inline "cic:/CoRN/algebra/Expon/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 inline "cic:/CoRN/algebra/Expon/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].
119 (*#* **Properties of [zexp]
120 %\begin{convention}% Let [R] be an ordered field.
125 Section Zexp_properties.
128 inline "cic:/CoRN/algebra/Expon/R.var".
130 inline "cic:/CoRN/algebra/Expon/zexp_zero.con".
133 Hint Resolve zexp_zero: algebra.
136 inline "cic:/CoRN/algebra/Expon/zexp_nexp.con".
139 Hint Resolve zexp_nexp: algebra.
142 inline "cic:/CoRN/algebra/Expon/zexp_inv_nexp.con".
145 Hint Resolve zexp_inv_nexp: algebra.
148 inline "cic:/CoRN/algebra/Expon/zexp_inv_nexp'.con".
151 Hint Resolve zexp_inv_nexp': algebra.
154 inline "cic:/CoRN/algebra/Expon/zexp_strext.con".
156 inline "cic:/CoRN/algebra/Expon/zexp_wd.con".
159 Hint Resolve zexp_wd: algebra_c.
162 inline "cic:/CoRN/algebra/Expon/zexp_plus1.con".
165 Hint Resolve zexp_plus1: algebra.
168 inline "cic:/CoRN/algebra/Expon/zexp_resp_ap_zero.con".
171 Hint Resolve zexp_resp_ap_zero: algebra.
174 inline "cic:/CoRN/algebra/Expon/zexp_inv.con".
177 Hint Resolve zexp_inv: algebra.
180 inline "cic:/CoRN/algebra/Expon/zexp_inv1.con".
183 Hint Resolve zexp_inv1: algebra.
186 inline "cic:/CoRN/algebra/Expon/zexp_plus.con".
189 Hint Resolve zexp_plus: algebra.
192 inline "cic:/CoRN/algebra/Expon/zexp_minus.con".
195 Hint Resolve zexp_minus: algebra.
198 inline "cic:/CoRN/algebra/Expon/one_zexp.con".
201 Hint Resolve one_zexp: algebra.
204 inline "cic:/CoRN/algebra/Expon/mult_zexp.con".
207 Hint Resolve mult_zexp: algebra.
210 inline "cic:/CoRN/algebra/Expon/zexp_mult.con".
213 Hint Resolve zexp_mult: algebra.
216 inline "cic:/CoRN/algebra/Expon/zexp_two.con".
219 Hint Resolve zexp_two: algebra.
222 inline "cic:/CoRN/algebra/Expon/inv_zexp_even.con".
225 Hint Resolve inv_zexp_even: algebra.
228 inline "cic:/CoRN/algebra/Expon/inv_zexp_two.con".
231 Hint Resolve inv_zexp_two: algebra.
234 inline "cic:/CoRN/algebra/Expon/inv_zexp_odd.con".
236 inline "cic:/CoRN/algebra/Expon/zexp_one.con".
239 Hint Resolve zexp_one: algebra.
242 inline "cic:/CoRN/algebra/Expon/zexp_funny.con".
245 Hint Resolve zexp_funny: algebra.
248 inline "cic:/CoRN/algebra/Expon/zexp_funny'.con".
251 Hint Resolve zexp_funny': algebra.
254 inline "cic:/CoRN/algebra/Expon/zexp_pos.con".
261 Hint Resolve nexp_resp_ap_zero zexp_zero zexp_nexp zexp_inv_nexp
262 zexp_inv_nexp' zexp_plus1 zexp_resp_ap_zero zexp_inv zexp_inv1 zexp_plus
263 zexp_minus one_zexp mult_zexp zexp_mult zexp_two inv_zexp_even inv_zexp_two
264 zexp_one zexp_funny zexp_funny': algebra.
268 Hint Resolve zexp_wd: algebra_c.
275 inline "cic:/CoRN/algebra/Expon/R.var".
277 inline "cic:/CoRN/algebra/Expon/root_unique.con".
279 inline "cic:/CoRN/algebra/Expon/root_one.con".