set "baseuri" "cic:/matita/CoRN-Decl/reals/Intervals".
+include "CoRN_notation.ma".
+
(* $Id: Intervals.v,v 1.10 2004/04/23 10:01:04 lcf Exp $ *)
-(* INCLUDE
-CSetoidInc
-*)
+include "algebra/CSetoidInc.ma".
-(* INCLUDE
-RealLists
-*)
+include "reals/RealLists.ma".
(* UNEXPORTED
Section Intervals.
require [a [<=] b], as we want to work only in nonempty intervals.
*)
-inline cic:/CoRN/reals/Intervals/compact.con.
+inline "cic:/CoRN/reals/Intervals/compact.con".
(*#*
%\begin{convention}% Let [a,b : IR] and [Hab : a [<=] b].
members of the interval [[Min(a,b),Max(a,b)]].
*)
-inline cic:/CoRN/reals/Intervals/a.var.
+inline "cic:/CoRN/reals/Intervals/a.var".
-inline cic:/CoRN/reals/Intervals/b.var.
+inline "cic:/CoRN/reals/Intervals/b.var".
-inline cic:/CoRN/reals/Intervals/Hab.var.
+inline "cic:/CoRN/reals/Intervals/Hab.var".
-inline cic:/CoRN/reals/Intervals/compact_inc_lft.con.
+inline "cic:/CoRN/reals/Intervals/compact_inc_lft.con".
-inline cic:/CoRN/reals/Intervals/compact_inc_rht.con.
+inline "cic:/CoRN/reals/Intervals/compact_inc_rht.con".
-inline cic:/CoRN/reals/Intervals/compact_Min_lft.con.
+inline "cic:/CoRN/reals/Intervals/compact_Min_lft.con".
-inline cic:/CoRN/reals/Intervals/compact_Min_rht.con.
+inline "cic:/CoRN/reals/Intervals/compact_Min_rht.con".
(*#*
As we will be interested in taking functions with domain in a compact
interval, we want this predicate to be well defined.
*)
-inline cic:/CoRN/reals/Intervals/compact_wd.con.
+inline "cic:/CoRN/reals/Intervals/compact_wd.con".
(*#*
Also, it will sometimes be necessary to rewrite the endpoints of an interval.
*)
-inline cic:/CoRN/reals/Intervals/compact_wd'.con.
+inline "cic:/CoRN/reals/Intervals/compact_wd'.con".
(*#*
As we identify subsets with predicates, inclusion is simply implication.
returns the restriction $F|_P$# of F to P#.
*)
-inline cic:/CoRN/reals/Intervals/Frestr.con.
+inline "cic:/CoRN/reals/Intervals/Frestr.con".
(* UNEXPORTED
End Intervals.
Section More_Intervals.
*)
-inline cic:/CoRN/reals/Intervals/included_refl'.con.
+inline "cic:/CoRN/reals/Intervals/included_refl'.con".
(*#* We prove some inclusions of compact intervals. *)
-inline cic:/CoRN/reals/Intervals/compact_map1.con.
+inline "cic:/CoRN/reals/Intervals/compact_map1.con".
-inline cic:/CoRN/reals/Intervals/compact_map2.con.
+inline "cic:/CoRN/reals/Intervals/compact_map2.con".
-inline cic:/CoRN/reals/Intervals/compact_map3.con.
+inline "cic:/CoRN/reals/Intervals/compact_map3.con".
(* UNEXPORTED
End More_Intervals.
Notice the use of lists for quantification.
*)
-inline cic:/CoRN/reals/Intervals/totally_bounded.con.
+inline "cic:/CoRN/reals/Intervals/totally_bounded.con".
(*#*
This definition is classically, but not constructively, equivalent to
and that we defined compacts as we did.
*)
-inline cic:/CoRN/reals/Intervals/compact_is_totally_bounded.con.
+inline "cic:/CoRN/reals/Intervals/compact_is_totally_bounded.con".
(*#*
Suprema and infima will be needed throughout; we define them here both
for arbitrary subsets of [IR] and for images of functions.
*)
-inline cic:/CoRN/reals/Intervals/set_glb_IR.con.
+inline "cic:/CoRN/reals/Intervals/set_glb_IR.con".
-inline cic:/CoRN/reals/Intervals/set_lub_IR.con.
+inline "cic:/CoRN/reals/Intervals/set_lub_IR.con".
-inline cic:/CoRN/reals/Intervals/fun_image.con.
+inline "cic:/CoRN/reals/Intervals/fun_image.con".
-inline cic:/CoRN/reals/Intervals/fun_glb_IR.con.
+inline "cic:/CoRN/reals/Intervals/fun_glb_IR.con".
-inline cic:/CoRN/reals/Intervals/fun_lub_IR.con.
+inline "cic:/CoRN/reals/Intervals/fun_lub_IR.con".
(* begin hide *)
-inline cic:/CoRN/reals/Intervals/aux_seq_lub.con.
+inline "cic:/CoRN/reals/Intervals/aux_seq_lub.con".
-inline cic:/CoRN/reals/Intervals/aux_seq_lub_prop.con.
+inline "cic:/CoRN/reals/Intervals/aux_seq_lub_prop.con".
(* end hide *)
The following are probably the most important results in this section.
*)
-inline cic:/CoRN/reals/Intervals/totally_bounded_has_lub.con.
+inline "cic:/CoRN/reals/Intervals/totally_bounded_has_lub.con".
(* begin hide *)
-inline cic:/CoRN/reals/Intervals/aux_seq_glb.con.
+inline "cic:/CoRN/reals/Intervals/aux_seq_glb.con".
-inline cic:/CoRN/reals/Intervals/aux_seq_glb_prop.con.
+inline "cic:/CoRN/reals/Intervals/aux_seq_glb_prop.con".
(* end hide *)
-inline cic:/CoRN/reals/Intervals/totally_bounded_has_glb.con.
+inline "cic:/CoRN/reals/Intervals/totally_bounded_has_glb.con".
(* UNEXPORTED
End Totally_Bounded.
The following characterization of inclusion can be very useful:
*)
-inline cic:/CoRN/reals/Intervals/included_compact.con.
+inline "cic:/CoRN/reals/Intervals/included_compact.con".
(*#*
At several points in our future development of a theory we will need
%\end{convention}%
*)
-inline cic:/CoRN/reals/Intervals/a.var.
+inline "cic:/CoRN/reals/Intervals/a.var".
-inline cic:/CoRN/reals/Intervals/b.var.
+inline "cic:/CoRN/reals/Intervals/b.var".
-inline cic:/CoRN/reals/Intervals/Hab.var.
+inline "cic:/CoRN/reals/Intervals/Hab.var".
(* begin hide *)
-inline cic:/CoRN/reals/Intervals/I.con.
+inline "cic:/CoRN/reals/Intervals/I.con".
(* end hide *)
-inline cic:/CoRN/reals/Intervals/Hab'.var.
+inline "cic:/CoRN/reals/Intervals/Hab'.var".
-inline cic:/CoRN/reals/Intervals/e.var.
+inline "cic:/CoRN/reals/Intervals/e.var".
-inline cic:/CoRN/reals/Intervals/He.var.
+inline "cic:/CoRN/reals/Intervals/He.var".
(*#*
We start by finding a natural number [n] such that [(b[-]a) [/] n [<] e].
*)
-inline cic:/CoRN/reals/Intervals/compact_nat.con.
+inline "cic:/CoRN/reals/Intervals/compact_nat.con".
(*#* Obviously such an [n] must be greater than zero.*)
-inline cic:/CoRN/reals/Intervals/pos_compact_nat.con.
+inline "cic:/CoRN/reals/Intervals/pos_compact_nat.con".
(*#*
We now define a sequence on [n] points in [[a,b]] by
prove that all of its points are really in that interval.
*)
-inline cic:/CoRN/reals/Intervals/compact_part.con.
+inline "cic:/CoRN/reals/Intervals/compact_part.con".
-inline cic:/CoRN/reals/Intervals/compact_part_hyp.con.
+inline "cic:/CoRN/reals/Intervals/compact_part_hyp.con".
(*#*
This sequence is strictly increasing and each two consecutive points
are apart by less than [e].*)
-inline cic:/CoRN/reals/Intervals/compact_less.con.
+inline "cic:/CoRN/reals/Intervals/compact_less.con".
-inline cic:/CoRN/reals/Intervals/compact_leEq.con.
+inline "cic:/CoRN/reals/Intervals/compact_leEq.con".
(*#* When we proceed to integration, this lemma will also be useful: *)
-inline cic:/CoRN/reals/Intervals/compact_partition_lemma.con.
+inline "cic:/CoRN/reals/Intervals/compact_partition_lemma.con".
(*#* The next lemma provides an upper bound for the distance between two points in an interval: *)
-inline cic:/CoRN/reals/Intervals/compact_elements.con.
+inline "cic:/CoRN/reals/Intervals/compact_elements.con".
(* UNEXPORTED
Opaque Min Max.
(*#* The following is a variation on the previous lemma: *)
-inline cic:/CoRN/reals/Intervals/compact_elements'.con.
+inline "cic:/CoRN/reals/Intervals/compact_elements'.con".
(*#* 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 cic:/CoRN/reals/Intervals/compact_bnd_AbsIR.con.
+inline "cic:/CoRN/reals/Intervals/compact_bnd_AbsIR.con".
(*#* 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 cic:/CoRN/reals/Intervals/included2_compact.con.
+inline "cic:/CoRN/reals/Intervals/included2_compact.con".
-inline cic:/CoRN/reals/Intervals/included3_compact.con.
+inline "cic:/CoRN/reals/Intervals/included3_compact.con".
(* UNEXPORTED
End Compact.