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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)]].
50 alias id "a" = "cic:/CoRN/reals/Intervals/Intervals/a.var".
52 alias id "b" = "cic:/CoRN/reals/Intervals/Intervals/b.var".
54 alias id "Hab" = "cic:/CoRN/reals/Intervals/Intervals/Hab.var".
56 inline procedural "cic:/CoRN/reals/Intervals/compact_inc_lft.con" as lemma.
58 inline procedural "cic:/CoRN/reals/Intervals/compact_inc_rht.con" as lemma.
60 inline procedural "cic:/CoRN/reals/Intervals/compact_Min_lft.con" as lemma.
62 inline procedural "cic:/CoRN/reals/Intervals/compact_Min_rht.con" as lemma.
65 As we will be interested in taking functions with domain in a compact
66 interval, we want this predicate to be well defined.
69 inline procedural "cic:/CoRN/reals/Intervals/compact_wd.con" as lemma.
72 Also, it will sometimes be necessary to rewrite the endpoints of an interval.
75 inline procedural "cic:/CoRN/reals/Intervals/compact_wd'.con" as lemma.
78 As we identify subsets with predicates, inclusion is simply implication.
82 Finally, we define a restriction operator that takes a function [F]
83 and a well defined predicate [P] included in the domain of [F] and
84 returns the restriction $F|_P$# of F to P#.
87 inline procedural "cic:/CoRN/reals/Intervals/Frestr.con" as definition.
94 Notation Compact := (compact _ _).
98 Implicit Arguments Frestr [F P].
102 Notation FRestr := (Frestr (compact_wd _ _ _)).
106 Section More_Intervals
109 inline procedural "cic:/CoRN/reals/Intervals/included_refl'.con" as lemma.
111 (*#* We prove some inclusions of compact intervals. *)
113 inline procedural "cic:/CoRN/reals/Intervals/compact_map1.con" as definition.
115 inline procedural "cic:/CoRN/reals/Intervals/compact_map2.con" as definition.
117 inline procedural "cic:/CoRN/reals/Intervals/compact_map3.con" as definition.
124 Hint Resolve included_refl' compact_map1 compact_map2 compact_map3 : included.
128 Section Totally_Bounded
131 (*#* ** Totally Bounded
133 Totally bounded sets will play an important role in what is
134 to come. The definition (equivalent to the classical one) states that
135 [P] is totally bounded iff
136 %\[\forall_{\varepsilon>0}\exists_{x_1,\ldots,x_n}\forall_{y\in P}
137 \exists_{1\leq i\leq n}|y-x_i|<\varepsilon\]%#∀e>0
138 ∃x<sub>1</sub>,...,x<sub>n</sub>∀y∈P∃
139 1≤i≤n.|y-x<sub>i</sub>|<e#.
141 Notice the use of lists for quantification.
144 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded.con" as definition.
147 This definition is classically, but not constructively, equivalent to
148 the definition of compact (if completeness is assumed); the next
149 result, classically equivalent to the Heine-Borel theorem, justifies
150 that we take the definition of totally bounded to be the relevant one
151 and that we defined compacts as we did.
154 inline procedural "cic:/CoRN/reals/Intervals/compact_is_totally_bounded.con" as lemma.
157 Suprema and infima will be needed throughout; we define them here both
158 for arbitrary subsets of [IR] and for images of functions.
161 inline procedural "cic:/CoRN/reals/Intervals/set_glb_IR.con" as definition.
163 inline procedural "cic:/CoRN/reals/Intervals/set_lub_IR.con" as definition.
165 inline procedural "cic:/CoRN/reals/Intervals/fun_image.con" as definition.
167 inline procedural "cic:/CoRN/reals/Intervals/fun_glb_IR.con" as definition.
169 inline procedural "cic:/CoRN/reals/Intervals/fun_lub_IR.con" as definition.
173 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub.con" "Totally_Bounded__" as definition.
175 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub_prop.con" "Totally_Bounded__" as definition.
180 The following are probably the most important results in this section.
183 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_lub.con" as lemma.
187 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb.con" "Totally_Bounded__" as definition.
189 inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb_prop.con" "Totally_Bounded__" as definition.
193 inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_glb.con" as lemma.
205 In this section we dwell a bit farther into the definition of compactness
206 and explore some of its properties.
208 The following characterization of inclusion can be very useful:
211 inline procedural "cic:/CoRN/reals/Intervals/included_compact.con" as lemma.
214 At several points in our future development of a theory we will need
215 to partition a compact interval in subintervals of length smaller than
216 some predefined value [eps]. Although this seems a
217 consequence of every compact interval being totally bounded, it is in
218 fact a stronger property. In this section we perform that
219 construction (requiring the endpoints of the interval to be distinct)
220 and prove some of its good properties.
222 %\begin{convention}% Let [a,b : IR], [Hab : (a [<=] b)] and denote by [I]
223 the compact interval [[a,b]]. Also assume that [a [<] b], and let [e] be
224 a positive real number.
228 alias id "a" = "cic:/CoRN/reals/Intervals/Compact/a.var".
230 alias id "b" = "cic:/CoRN/reals/Intervals/Compact/b.var".
232 alias id "Hab" = "cic:/CoRN/reals/Intervals/Compact/Hab.var".
236 inline procedural "cic:/CoRN/reals/Intervals/Compact/I.con" "Compact__" as definition.
240 alias id "Hab'" = "cic:/CoRN/reals/Intervals/Compact/Hab'.var".
242 alias id "e" = "cic:/CoRN/reals/Intervals/Compact/e.var".
244 alias id "He" = "cic:/CoRN/reals/Intervals/Compact/He.var".
247 We start by finding a natural number [n] such that [(b[-]a) [/] n [<] e].
250 inline procedural "cic:/CoRN/reals/Intervals/compact_nat.con" as definition.
252 (*#* Obviously such an [n] must be greater than zero.*)
254 inline procedural "cic:/CoRN/reals/Intervals/pos_compact_nat.con" as lemma.
257 We now define a sequence on [n] points in [[a,b]] by
258 [x_i [=] Min(a,b) [+]i[*] (b[-]a) [/]n] and
259 prove that all of its points are really in that interval.
262 inline procedural "cic:/CoRN/reals/Intervals/compact_part.con" as definition.
264 inline procedural "cic:/CoRN/reals/Intervals/compact_part_hyp.con" as lemma.
267 This sequence is strictly increasing and each two consecutive points
268 are apart by less than [e].*)
270 inline procedural "cic:/CoRN/reals/Intervals/compact_less.con" as lemma.
272 inline procedural "cic:/CoRN/reals/Intervals/compact_leEq.con" as lemma.
274 (*#* When we proceed to integration, this lemma will also be useful: *)
276 inline procedural "cic:/CoRN/reals/Intervals/compact_partition_lemma.con" as lemma.
278 (*#* The next lemma provides an upper bound for the distance between two points in an interval: *)
280 inline procedural "cic:/CoRN/reals/Intervals/compact_elements.con" as lemma.
286 (*#* The following is a variation on the previous lemma: *)
288 inline procedural "cic:/CoRN/reals/Intervals/compact_elements'.con" as lemma.
290 (*#* The following lemma is a bit more specific: it shows that we can
291 also estimate the distance from the center of a compact interval to
292 any of its points. *)
294 inline procedural "cic:/CoRN/reals/Intervals/compact_bnd_AbsIR.con" as lemma.
296 (*#* Finally, two more useful lemmas to prove inclusion of compact
297 intervals. They will be necessary for the definition and proof of the
298 elementary properties of the integral. *)
300 inline procedural "cic:/CoRN/reals/Intervals/included2_compact.con" as lemma.
302 inline procedural "cic:/CoRN/reals/Intervals/included3_compact.con" as lemma.
309 Hint Resolve included_compact included2_compact included3_compact : included.