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 *********************)
19 (*#**********************************************************************)
21 (* v * The Coq Proof Assistant / The Coq Development Team *)
23 (* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
25 (* \VV/ *************************************************************)
27 (* // * This file is distributed under the terms of the *)
29 (* * GNU Lesser General Public License Version 2.1 *)
31 (*#**********************************************************************)
33 (*i $Id: Rfunctions.v,v 1.31 2004/03/12 10:15:44 mohring Exp $ i*)
35 (*i Some properties about pow and sum have been made with John Harrison i*)
37 (*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*)
39 (*#*******************************************************)
41 (*#* Definition of the sum functions *)
45 (*#*******************************************************)
47 include "Reals/Rbase.ma".
49 include "Reals/R_Ifp.ma".
51 include "Reals/Rbasic_fun.ma".
53 include "Reals/R_sqr.ma".
55 include "Reals/SplitAbsolu.ma".
57 include "Reals/SplitRmult.ma".
59 include "Reals/ArithProp.ma".
61 include "ZArith/Zpower.ma".
64 Open Local Scope nat_scope.
68 Open Local Scope R_scope.
71 (*#******************************)
73 (*#* Lemmas about factorial *)
75 (*#******************************)
79 inline procedural "cic:/Coq/Reals/Rfunctions/INR_fact_neq_0.con" as lemma.
83 inline procedural "cic:/Coq/Reals/Rfunctions/fact_simpl.con" as lemma.
87 inline procedural "cic:/Coq/Reals/Rfunctions/simpl_fact.con" as lemma.
89 (*#******************************)
93 (*#******************************)
97 inline procedural "cic:/Coq/Reals/Rfunctions/pow.con" as definition.
100 Infix "^" := pow : R_scope.
103 inline procedural "cic:/Coq/Reals/Rfunctions/pow_O.con" as lemma.
105 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1.con" as lemma.
107 inline procedural "cic:/Coq/Reals/Rfunctions/pow_add.con" as lemma.
109 inline procedural "cic:/Coq/Reals/Rfunctions/pow_nonzero.con" as lemma.
112 Hint Resolve pow_O pow_1 pow_add pow_nonzero: real.
115 inline procedural "cic:/Coq/Reals/Rfunctions/pow_RN_plus.con" as lemma.
117 inline procedural "cic:/Coq/Reals/Rfunctions/pow_lt.con" as lemma.
120 Hint Resolve pow_lt: real.
123 inline procedural "cic:/Coq/Reals/Rfunctions/Rlt_pow_R1.con" as lemma.
126 Hint Resolve Rlt_pow_R1: real.
129 inline procedural "cic:/Coq/Reals/Rfunctions/Rlt_pow.con" as lemma.
132 Hint Resolve Rlt_pow: real.
137 inline procedural "cic:/Coq/Reals/Rfunctions/tech_pow_Rmult.con" as lemma.
141 inline procedural "cic:/Coq/Reals/Rfunctions/tech_pow_Rplus.con" as lemma.
143 inline procedural "cic:/Coq/Reals/Rfunctions/poly.con" as lemma.
145 inline procedural "cic:/Coq/Reals/Rfunctions/Power_monotonic.con" as lemma.
147 inline procedural "cic:/Coq/Reals/Rfunctions/RPow_abs.con" as lemma.
149 inline procedural "cic:/Coq/Reals/Rfunctions/Pow_x_infinity.con" as lemma.
151 inline procedural "cic:/Coq/Reals/Rfunctions/pow_ne_zero.con" as lemma.
153 inline procedural "cic:/Coq/Reals/Rfunctions/Rinv_pow.con" as lemma.
155 inline procedural "cic:/Coq/Reals/Rfunctions/pow_lt_1_zero.con" as lemma.
157 inline procedural "cic:/Coq/Reals/Rfunctions/pow_R1.con" as lemma.
159 inline procedural "cic:/Coq/Reals/Rfunctions/pow_Rsqr.con" as lemma.
161 inline procedural "cic:/Coq/Reals/Rfunctions/pow_le.con" as lemma.
165 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_even.con" as lemma.
169 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_odd.con" as lemma.
173 inline procedural "cic:/Coq/Reals/Rfunctions/pow_1_abs.con" as lemma.
175 inline procedural "cic:/Coq/Reals/Rfunctions/pow_mult.con" as lemma.
177 inline procedural "cic:/Coq/Reals/Rfunctions/pow_incr.con" as lemma.
179 inline procedural "cic:/Coq/Reals/Rfunctions/pow_R1_Rle.con" as lemma.
181 inline procedural "cic:/Coq/Reals/Rfunctions/Rle_pow.con" as lemma.
183 inline procedural "cic:/Coq/Reals/Rfunctions/pow1.con" as lemma.
185 inline procedural "cic:/Coq/Reals/Rfunctions/pow_Rabs.con" as lemma.
187 inline procedural "cic:/Coq/Reals/Rfunctions/pow_maj_Rabs.con" as lemma.
189 (*#******************************)
193 (*#******************************)
195 (*i Due to L.Thery i*)
199 generalize (refl_equal name); pattern name at -1 in |- *; case name.
202 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ.con" as definition.
205 Infix Local "^Z" := powerRZ (at level 30, right associativity) : R_scope.
208 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_NR0.con" as lemma.
210 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_O.con" as lemma.
212 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_1.con" as lemma.
214 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_NOR.con" as lemma.
216 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_add.con" as lemma.
219 Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.
222 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_nat_powerRZ.con" as lemma.
224 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_lt.con" as lemma.
227 Hint Resolve powerRZ_lt: real.
230 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_le.con" as lemma.
233 Hint Resolve powerRZ_le: real.
236 inline procedural "cic:/Coq/Reals/Rfunctions/Zpower_nat_powerRZ_absolu.con" as lemma.
238 inline procedural "cic:/Coq/Reals/Rfunctions/powerRZ_R1.con" as lemma.
240 (*#******************************)
242 (* For easy interface *)
244 (*#******************************)
246 (* decimal_exp r z is defined as r 10^z *)
248 inline procedural "cic:/Coq/Reals/Rfunctions/decimal_exp.con" as definition.
250 (*#******************************)
252 (*#* Sum of n first naturals *)
254 (*#******************************)
258 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_f_O.con" as definition.
262 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_f.con" as definition.
266 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat_O.con" as definition.
270 inline procedural "cic:/Coq/Reals/Rfunctions/sum_nat.con" as definition.
272 (*#******************************)
276 (*#******************************)
280 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f_R0.con" as definition.
284 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f.con" as definition.
286 inline procedural "cic:/Coq/Reals/Rfunctions/GP_finite.con" as lemma.
288 inline procedural "cic:/Coq/Reals/Rfunctions/sum_f_R0_triangle.con" as lemma.
290 (*#******************************)
294 (*#******************************)
298 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist.con" as definition.
302 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_pos.con" as lemma.
306 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_sym.con" as lemma.
310 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_refl.con" as lemma.
312 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_eq.con" as lemma.
316 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_tri.con" as lemma.
320 inline procedural "cic:/Coq/Reals/Rfunctions/R_dist_plus.con" as lemma.
322 (*#******************************)
326 (*#******************************)
330 inline procedural "cic:/Coq/Reals/Rfunctions/infinit_sum.con" as definition.