(**************************************************************************)
(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
(* ||I|| Developers: *)
(* ||T|| The HELM team. *)
(* ||A|| http://helm.cs.unibo.it *)
(* \ / *)
(* \ / This file is distributed under the terms of the *)
(* v GNU General Public License Version 2 *)
(* *)
(**************************************************************************)
(* This file was automatically generated: do not edit *********************)
include "CoRN.ma".
(* $Id: Intervals.v,v 1.10 2004/04/23 10:01:04 lcf Exp $ *)
include "algebra/CSetoidInc.ma".
include "reals/RealLists.ma".
(* UNEXPORTED
Section Intervals
*)
(*#* * Intervals
In this section we define (compact) intervals of the real line and
some useful functions to work with them.
** Definitions
We start by defining the compact interval [[a,b]] as being the set of
points less or equal than [b] and greater or equal than [a]. We
require [a [<=] b], as we want to work only in nonempty intervals.
*)
inline procedural "cic:/CoRN/reals/Intervals/compact.con" as definition.
(*#*
%\begin{convention}% Let [a,b : IR] and [Hab : a [<=] b].
%\end{convention}%
As expected, both [a] and [b] are members of [[a,b]]. Also they are
members of the interval [[Min(a,b),Max(a,b)]].
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Intervals/a.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Intervals/b.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Intervals/Hab.var
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_inc_lft.con" as lemma.
inline procedural "cic:/CoRN/reals/Intervals/compact_inc_rht.con" as lemma.
inline procedural "cic:/CoRN/reals/Intervals/compact_Min_lft.con" as lemma.
inline procedural "cic:/CoRN/reals/Intervals/compact_Min_rht.con" as lemma.
(*#*
As we will be interested in taking functions with domain in a compact
interval, we want this predicate to be well defined.
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_wd.con" as lemma.
(*#*
Also, it will sometimes be necessary to rewrite the endpoints of an interval.
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_wd'.con" as lemma.
(*#*
As we identify subsets with predicates, inclusion is simply implication.
*)
(*#*
Finally, we define a restriction operator that takes a function [F]
and a well defined predicate [P] included in the domain of [F] and
returns the restriction $F|_P$# of F to P#.
*)
inline procedural "cic:/CoRN/reals/Intervals/Frestr.con" as definition.
(* UNEXPORTED
End Intervals
*)
(* NOTATION
Notation Compact := (compact _ _).
*)
(* UNEXPORTED
Implicit Arguments Frestr [F P].
*)
(* NOTATION
Notation FRestr := (Frestr (compact_wd _ _ _)).
*)
(* UNEXPORTED
Section More_Intervals
*)
inline procedural "cic:/CoRN/reals/Intervals/included_refl'.con" as lemma.
(*#* We prove some inclusions of compact intervals. *)
inline procedural "cic:/CoRN/reals/Intervals/compact_map1.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/compact_map2.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/compact_map3.con" as definition.
(* UNEXPORTED
End More_Intervals
*)
(* UNEXPORTED
Hint Resolve included_refl' compact_map1 compact_map2 compact_map3 : included.
*)
(* UNEXPORTED
Section Totally_Bounded
*)
(*#* ** Totally Bounded
Totally bounded sets will play an important role in what is
to come. The definition (equivalent to the classical one) states that
[P] is totally bounded iff
%\[\forall_{\varepsilon>0}\exists_{x_1,\ldots,x_n}\forall_{y\in P}
\exists_{1\leq i\leq n}|y-x_i|<\varepsilon\]%#∀e>0
∃x1,...,xn∀y∈P∃
1≤i≤n.|y-xi|<e#.
Notice the use of lists for quantification.
*)
inline procedural "cic:/CoRN/reals/Intervals/totally_bounded.con" as definition.
(*#*
This definition is classically, but not constructively, equivalent to
the definition of compact (if completeness is assumed); the next
result, classically equivalent to the Heine-Borel theorem, justifies
that we take the definition of totally bounded to be the relevant one
and that we defined compacts as we did.
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_is_totally_bounded.con" as lemma.
(*#*
Suprema and infima will be needed throughout; we define them here both
for arbitrary subsets of [IR] and for images of functions.
*)
inline procedural "cic:/CoRN/reals/Intervals/set_glb_IR.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/set_lub_IR.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/fun_image.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/fun_glb_IR.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/fun_lub_IR.con" as definition.
(* begin hide *)
inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub.con" "Totally_Bounded__" as definition.
inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_lub_prop.con" "Totally_Bounded__" as definition.
(* end hide *)
(*#*
The following are probably the most important results in this section.
*)
inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_lub.con" as lemma.
(* begin hide *)
inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb.con" "Totally_Bounded__" as definition.
inline procedural "cic:/CoRN/reals/Intervals/Totally_Bounded/aux_seq_glb_prop.con" "Totally_Bounded__" as definition.
(* end hide *)
inline procedural "cic:/CoRN/reals/Intervals/totally_bounded_has_glb.con" as lemma.
(* UNEXPORTED
End Totally_Bounded
*)
(* UNEXPORTED
Section Compact
*)
(*#* ** Compact sets
In this section we dwell a bit farther into the definition of compactness
and explore some of its properties.
The following characterization of inclusion can be very useful:
*)
inline procedural "cic:/CoRN/reals/Intervals/included_compact.con" as lemma.
(*#*
At several points in our future development of a theory we will need
to partition a compact interval in subintervals of length smaller than
some predefined value [eps]. Although this seems a
consequence of every compact interval being totally bounded, it is in
fact a stronger property. In this section we perform that
construction (requiring the endpoints of the interval to be distinct)
and prove some of its good properties.
%\begin{convention}% Let [a,b : IR], [Hab : (a [<=] b)] and denote by [I]
the compact interval [[a,b]]. Also assume that [a [<] b], and let [e] be
a positive real number.
%\end{convention}%
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/a.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/b.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/Hab.var
*)
(* begin hide *)
inline procedural "cic:/CoRN/reals/Intervals/Compact/I.con" "Compact__" as definition.
(* end hide *)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/Hab'.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/e.var
*)
(* UNEXPORTED
cic:/CoRN/reals/Intervals/Compact/He.var
*)
(*#*
We start by finding a natural number [n] such that [(b[-]a) [/] n [<] e].
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_nat.con" as definition.
(*#* Obviously such an [n] must be greater than zero.*)
inline procedural "cic:/CoRN/reals/Intervals/pos_compact_nat.con" as lemma.
(*#*
We now define a sequence on [n] points in [[a,b]] by
[x_i [=] Min(a,b) [+]i[*] (b[-]a) [/]n] and
prove that all of its points are really in that interval.
*)
inline procedural "cic:/CoRN/reals/Intervals/compact_part.con" as definition.
inline procedural "cic:/CoRN/reals/Intervals/compact_part_hyp.con" as lemma.
(*#*
This sequence is strictly increasing and each two consecutive points
are apart by less than [e].*)
inline procedural "cic:/CoRN/reals/Intervals/compact_less.con" as lemma.
inline procedural "cic:/CoRN/reals/Intervals/compact_leEq.con" as lemma.
(*#* When we proceed to integration, this lemma will also be useful: *)
inline procedural "cic:/CoRN/reals/Intervals/compact_partition_lemma.con" as lemma.
(*#* The next lemma provides an upper bound for the distance between two points in an interval: *)
inline procedural "cic:/CoRN/reals/Intervals/compact_elements.con" as lemma.
(* UNEXPORTED
Opaque Min Max.
*)
(*#* The following is a variation on the previous lemma: *)
inline procedural "cic:/CoRN/reals/Intervals/compact_elements'.con" as lemma.
(*#* The following lemma is a bit more specific: it shows that we can
also estimate the distance from the center of a compact interval to
any of its points. *)
inline procedural "cic:/CoRN/reals/Intervals/compact_bnd_AbsIR.con" as lemma.
(*#* Finally, two more useful lemmas to prove inclusion of compact
intervals. They will be necessary for the definition and proof of the
elementary properties of the integral. *)
inline procedural "cic:/CoRN/reals/Intervals/included2_compact.con" as lemma.
inline procedural "cic:/CoRN/reals/Intervals/included3_compact.con" as lemma.
(* UNEXPORTED
End Compact
*)
(* UNEXPORTED
Hint Resolve included_compact included2_compact included3_compact : included.
*)