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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: Intervals.v,v 1.10 2004/04/23 10:01:04 lcf Exp $ *)
21 include "algebra/CSetoidInc.ma".
23 include "reals/RealLists.ma".
30 In this section we define (compact) intervals of the real line and
31 some useful functions to work with them.
35 We start by defining the compact interval [[a,b]] as being the set of
36 points less or equal than [b] and greater or equal than [a]. We
37 require [a [<=] b], as we want to work only in nonempty intervals.
40 inline procedural "cic:/CoRN/reals/Intervals/compact.con" as definition.
43 %\begin{convention}% Let [a,b : IR] and [Hab : a [<=] b].
46 As expected, both [a] and [b] are members of [[a,b]]. Also they are
47 members of the interval [[Min(a,b),Max(a,b)]].
51 cic:/CoRN/reals/Intervals/Intervals/a.var
55 cic:/CoRN/reals/Intervals/Intervals/b.var
59 cic:/CoRN/reals/Intervals/Intervals/Hab.var
62 inline procedural "cic:/CoRN/reals/Intervals/compact_inc_lft.con" as lemma.
64 inline procedural "cic:/CoRN/reals/Intervals/compact_inc_rht.con" as lemma.
66 inline procedural "cic:/CoRN/reals/Intervals/compact_Min_lft.con" as lemma.
68 inline procedural "cic:/CoRN/reals/Intervals/compact_Min_rht.con" as lemma.
71 As we will be interested in taking functions with domain in a compact
72 interval, we want this predicate to be well defined.
75 inline procedural "cic:/CoRN/reals/Intervals/compact_wd.con" as lemma.
78 Also, it will sometimes be necessary to rewrite the endpoints of an interval.
81 inline procedural "cic:/CoRN/reals/Intervals/compact_wd'.con" as lemma.
84 As we identify subsets with predicates, inclusion is simply implication.
88 Finally, we define a restriction operator that takes a function [F]
89 and a well defined predicate [P] included in the domain of [F] and
90 returns the restriction $F|_P$# of F to P#.
93 inline procedural "cic:/CoRN/reals/Intervals/Frestr.con" as definition.
100 Notation Compact := (compact _ _).
104 Implicit Arguments Frestr [F P].
108 Notation FRestr := (Frestr (compact_wd _ _ _)).
112 Section More_Intervals
115 inline procedural "cic:/CoRN/reals/Intervals/included_refl'.con" as lemma.
117 (*#* We prove some inclusions of compact intervals. *)
119 inline procedural "cic:/CoRN/reals/Intervals/compact_map1.con" as definition.
121 inline procedural "cic:/CoRN/reals/Intervals/compact_map2.con" as definition.
123 inline procedural "cic:/CoRN/reals/Intervals/compact_map3.con" as definition.
130 Hint Resolve included_refl' compact_map1 compact_map2 compact_map3 : included.
134 Section Totally_Bounded
137 (*#* ** Totally Bounded
139 Totally bounded sets will play an important role in what is
140 to come. The definition (equivalent to the classical one) states that
141 [P] is totally bounded iff
142 %\[\forall_{\varepsilon>0}\exists_{x_1,\ldots,x_n}\forall_{y\in P}
143 \exists_{1\leq i\leq n}|y-x_i|<\varepsilon\]%#∀e>0
144 ∃x<sub>1</sub>,...,x<sub>n</sub>∀y∈P∃
145 1≤i≤n.|y-x<sub>i</sub>|<e#.
147 Notice the use of lists for quantification.
150 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded.con" as definition.
153 This definition is classically, but not constructively, equivalent to
154 the definition of compact (if completeness is assumed); the next
155 result, classically equivalent to the Heine-Borel theorem, justifies
156 that we take the definition of totally bounded to be the relevant one
157 and that we defined compacts as we did.
160 inline procedural "cic:/CoRN/reals/Intervals/compact_is_totally_bounded.con" as lemma.
163 Suprema and infima will be needed throughout; we define them here both
164 for arbitrary subsets of [IR] and for images of functions.
167 inline procedural "cic:/CoRN/reals/Intervals/set_glb_IR.con" as definition.
169 inline procedural "cic:/CoRN/reals/Intervals/set_lub_IR.con" as definition.
171 inline procedural "cic:/CoRN/reals/Intervals/fun_image.con" as definition.
173 inline procedural "cic:/CoRN/reals/Intervals/fun_glb_IR.con" as definition.
175 inline procedural "cic:/CoRN/reals/Intervals/fun_lub_IR.con" as definition.
179 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub.con" "Totally_Bounded__" as definition.
181 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub_prop.con" "Totally_Bounded__" as definition.
186 The following are probably the most important results in this section.
189 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_lub.con" as lemma.
193 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb.con" "Totally_Bounded__" as definition.
195 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb_prop.con" "Totally_Bounded__" as definition.
199 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_glb.con" as lemma.
211 In this section we dwell a bit farther into the definition of compactness
212 and explore some of its properties.
214 The following characterization of inclusion can be very useful:
217 inline procedural "cic:/CoRN/reals/Intervals/included_compact.con" as lemma.
220 At several points in our future development of a theory we will need
221 to partition a compact interval in subintervals of length smaller than
222 some predefined value [eps]. Although this seems a
223 consequence of every compact interval being totally bounded, it is in
224 fact a stronger property. In this section we perform that
225 construction (requiring the endpoints of the interval to be distinct)
226 and prove some of its good properties.
228 %\begin{convention}% Let [a,b : IR], [Hab : (a [<=] b)] and denote by [I]
229 the compact interval [[a,b]]. Also assume that [a [<] b], and let [e] be
230 a positive real number.
235 cic:/CoRN/reals/Intervals/Compact/a.var
239 cic:/CoRN/reals/Intervals/Compact/b.var
243 cic:/CoRN/reals/Intervals/Compact/Hab.var
248 inline procedural "cic:/CoRN/reals/Intervals/Compact/I.con" "Compact__" as definition.
253 cic:/CoRN/reals/Intervals/Compact/Hab'.var
257 cic:/CoRN/reals/Intervals/Compact/e.var
261 cic:/CoRN/reals/Intervals/Compact/He.var
265 We start by finding a natural number [n] such that [(b[-]a) [/] n [<] e].
268 inline procedural "cic:/CoRN/reals/Intervals/compact_nat.con" as definition.
270 (*#* Obviously such an [n] must be greater than zero.*)
272 inline procedural "cic:/CoRN/reals/Intervals/pos_compact_nat.con" as lemma.
275 We now define a sequence on [n] points in [[a,b]] by
276 [x_i [=] Min(a,b) [+]i[*] (b[-]a) [/]n] and
277 prove that all of its points are really in that interval.
280 inline procedural "cic:/CoRN/reals/Intervals/compact_part.con" as definition.
282 inline procedural "cic:/CoRN/reals/Intervals/compact_part_hyp.con" as lemma.
285 This sequence is strictly increasing and each two consecutive points
286 are apart by less than [e].*)
288 inline procedural "cic:/CoRN/reals/Intervals/compact_less.con" as lemma.
290 inline procedural "cic:/CoRN/reals/Intervals/compact_leEq.con" as lemma.
292 (*#* When we proceed to integration, this lemma will also be useful: *)
294 inline procedural "cic:/CoRN/reals/Intervals/compact_partition_lemma.con" as lemma.
296 (*#* The next lemma provides an upper bound for the distance between two points in an interval: *)
298 inline procedural "cic:/CoRN/reals/Intervals/compact_elements.con" as lemma.
304 (*#* The following is a variation on the previous lemma: *)
306 inline procedural "cic:/CoRN/reals/Intervals/compact_elements'.con" as lemma.
308 (*#* The following lemma is a bit more specific: it shows that we can
309 also estimate the distance from the center of a compact interval to
310 any of its points. *)
312 inline procedural "cic:/CoRN/reals/Intervals/compact_bnd_AbsIR.con" as lemma.
314 (*#* Finally, two more useful lemmas to prove inclusion of compact
315 intervals. They will be necessary for the definition and proof of the
316 elementary properties of the integral. *)
318 inline procedural "cic:/CoRN/reals/Intervals/included2_compact.con" as lemma.
320 inline procedural "cic:/CoRN/reals/Intervals/included3_compact.con" as lemma.
327 Hint Resolve included_compact included2_compact included3_compact : included.