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 (* $Id: IVT.v,v 1.5 2004/04/23 10:01:04 lcf Exp $ *)
21 include "reals/CPoly_Contin.ma".
24 Section Nested_Intervals
27 (*#* * Intermediate Value Theorem
31 %\begin{convention}% Let [a,b:nat->IR] be sequences such that:
34 - [forall (i:nat), (a i) [<] (b i)];
35 - for every positive real number [eps], there is an [i] such that [(b i) [<] (a i) [+]eps].
41 cic:/CoRN/reals/IVT/Nested_Intervals/a.var
45 cic:/CoRN/reals/IVT/Nested_Intervals/b.var
49 cic:/CoRN/reals/IVT/Nested_Intervals/a_mon.var
53 cic:/CoRN/reals/IVT/Nested_Intervals/b_mon.var
57 cic:/CoRN/reals/IVT/Nested_Intervals/a_b.var
61 cic:/CoRN/reals/IVT/Nested_Intervals/b_a.var
64 inline procedural "cic:/CoRN/reals/IVT/a_mon'.con" as lemma.
66 inline procedural "cic:/CoRN/reals/IVT/b_mon'.con" as lemma.
68 inline procedural "cic:/CoRN/reals/IVT/a_b'.con" as lemma.
70 inline procedural "cic:/CoRN/reals/IVT/intervals_cauchy.con" as lemma.
74 inline procedural "cic:/CoRN/reals/IVT/Nested_Intervals/a'.con" "Nested_Intervals__" as definition.
78 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_limit.con" as lemma.
80 (*#* %\begin{convention}% Let [f] be a continuous real function.
85 cic:/CoRN/reals/IVT/Nested_Intervals/f.var
89 cic:/CoRN/reals/IVT/Nested_Intervals/f_contin.var
92 inline procedural "cic:/CoRN/reals/IVT/f_contin_pos.con" as lemma.
94 inline procedural "cic:/CoRN/reals/IVT/f_contin_neg.con" as lemma.
96 (*#* Assume also that [forall i, f (a i) [<=] Zero [<=] f (b i)]. *)
99 cic:/CoRN/reals/IVT/Nested_Intervals/f_a.var
103 cic:/CoRN/reals/IVT/Nested_Intervals/f_b.var
106 inline procedural "cic:/CoRN/reals/IVT/Cnested_intervals_zero.con" as lemma.
116 (*#* ** Bissections *)
119 cic:/CoRN/reals/IVT/Bisection/f.var
123 cic:/CoRN/reals/IVT/Bisection/f_apzero_interval.var
127 cic:/CoRN/reals/IVT/Bisection/a.var
131 cic:/CoRN/reals/IVT/Bisection/b.var
135 cic:/CoRN/reals/IVT/Bisection/a_b.var
139 cic:/CoRN/reals/IVT/Bisection/f_a.var
143 cic:/CoRN/reals/IVT/Bisection/f_b.var
147 %\begin{convention}% Let [Small] denote [Two[/]ThreeNZ], [lft] be [(Two[*]a[+]b) [/]ThreeNZ] and [rht] be [(a[+]Two[*]b) [/]ThreeNZ].
153 inline procedural "cic:/CoRN/reals/IVT/Bisection/Small.con" "Bisection__" as definition.
155 inline procedural "cic:/CoRN/reals/IVT/Bisection/lft.con" "Bisection__" as definition.
157 inline procedural "cic:/CoRN/reals/IVT/Bisection/rht.con" "Bisection__" as definition.
161 inline procedural "cic:/CoRN/reals/IVT/a_lft.con" as lemma.
163 inline procedural "cic:/CoRN/reals/IVT/rht_b.con" as lemma.
165 inline procedural "cic:/CoRN/reals/IVT/lft_rht.con" as lemma.
167 inline procedural "cic:/CoRN/reals/IVT/smaller_lft.con" as lemma.
169 inline procedural "cic:/CoRN/reals/IVT/smaller_rht.con" as lemma.
172 Hint Resolve smaller_lft smaller_rht: algebra.
175 inline procedural "cic:/CoRN/reals/IVT/Cbisect'.con" as lemma.
182 Section Bisect_Interval
186 cic:/CoRN/reals/IVT/Bisect_Interval/f.var
190 cic:/CoRN/reals/IVT/Bisect_Interval/C_f_apzero_interval.var
195 inline procedural "cic:/CoRN/reals/IVT/Bisect_Interval/Small.con" "Bisect_Interval__" as definition.
199 inline procedural "cic:/CoRN/reals/IVT/bisect_interval.ind".
201 inline procedural "cic:/CoRN/reals/IVT/Cbisect_exists.con" as lemma.
203 inline procedural "cic:/CoRN/reals/IVT/bisect.con" as definition.
205 inline procedural "cic:/CoRN/reals/IVT/bisect_prop.con" as lemma.
215 (*#* ** IVT for operations
216 Same conventions as before.
220 cic:/CoRN/reals/IVT/IVT_Op/f.var
224 cic:/CoRN/reals/IVT/IVT_Op/f_contin.var
228 cic:/CoRN/reals/IVT/IVT_Op/f_apzero_interval.var
232 cic:/CoRN/reals/IVT/IVT_Op/a.var
236 cic:/CoRN/reals/IVT/IVT_Op/b.var
240 cic:/CoRN/reals/IVT/IVT_Op/a_b.var
244 cic:/CoRN/reals/IVT/IVT_Op/f_a.var
248 cic:/CoRN/reals/IVT/IVT_Op/f_b.var
253 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/Small.con" "IVT_Op__" as definition.
257 inline procedural "cic:/CoRN/reals/IVT/interval_sequence.con" as definition.
259 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/a_.con" "IVT_Op__" as definition.
261 inline procedural "cic:/CoRN/reals/IVT/IVT_Op/b_.con" "IVT_Op__" as definition.
263 inline procedural "cic:/CoRN/reals/IVT/intervals_smaller.con" as lemma.
265 inline procedural "cic:/CoRN/reals/IVT/intervals_small''.con" as lemma.
267 inline procedural "cic:/CoRN/reals/IVT/intervals_small'.con" as lemma.
269 inline procedural "cic:/CoRN/reals/IVT/intervals_small.con" as lemma.
271 inline procedural "cic:/CoRN/reals/IVT/Civt_op.con" as lemma.
281 (*#* ** IVT for polynomials *)
283 inline procedural "cic:/CoRN/reals/IVT/Civt_poly.con" as lemma.