--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/CoRN-Decl/ftc/Partitions".
+
+include "CoRN.ma".
+
+(* $Id: Partitions.v,v 1.7 2004/04/23 10:01:00 lcf Exp $ *)
+
+include "ftc/Continuity.ma".
+
+(*#* printing Partition_Sum %\ensuremath{\sum_P}% #∑<sub>P</sub># *)
+
+(* UNEXPORTED
+Section Definitions
+*)
+
+(*#* *Partitions
+
+We now begin to lay the way for the definition of Riemann integral. This will
+be done through the definition of a sequence of
+sums that is proved to be convergent; in order to do that, we first
+need to do a bit of work on partitions.
+
+**Definitions
+
+A partition is defined as a record type. For each compact interval [[a,b]]
+and each natural number [n], a partition of [[a,b]] with [n+1] points is a
+choice of real numbers [a [=] a0 [<=] a1 [<=] an [=] b]; the following
+specification works as follows:
+ - [Pts] is the function that chooses the points (it is declared as a
+coercion);
+ - [prf1] states that [Pts] is a setoid function;
+ - [prf2] states that the points are ordered;
+ - [start] requires that [a0 [=] a] and
+ - [finish] requires that [an [=] b].
+
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Partition.ind".
+
+coercion cic:/matita/CoRN-Decl/ftc/Partitions/Pts.con 0 (* compounds *).
+
+(*#* Two immediate consequences of this are that [ai [<=] aj] whenever
+[i < j] and that [ai] is in [[a,b]] for all [i].
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Partition_mon.con".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_in_compact.con".
+
+(*#*
+Each partition of [[a,b]] implies a partition of the interval
+$[a,a_{n-1}]$#[a,a<sub>n-1</sub>]#. This partition will play an
+important role in much of our future work, so we take some care to
+define it.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/part_pred_lemma.con".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_Dom.con".
+
+(*#*
+The mesh of a partition is the greatest distance between two
+consecutive points. For convenience's sake we also define the dual
+concept, which is very helpful in some situations. In order to do
+this, we begin by building a list with all the distances between
+consecutive points; next we only need to extract the maximum and the
+minimum of this list. Notice that this list is nonempty except in the
+case when [a [=] b] and [n = 0]; this means that the convention we took
+of defining the minimum and maximum of the empty list to be [0] actually
+helps us in this case.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Part_Mesh_List.con".
+
+inline "cic:/CoRN/ftc/Partitions/Mesh.con".
+
+inline "cic:/CoRN/ftc/Partitions/AntiMesh.con".
+
+(*#*
+Even partitions (partitions where all the points are evenly spaced)
+will also play a central role in our work; the first two lemmas are
+presented simply to make the definition of even partition lighter.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/even_part_1.con".
+
+inline "cic:/CoRN/ftc/Partitions/even_part_2.con".
+
+inline "cic:/CoRN/ftc/Partitions/Even_Partition.con".
+
+(* UNEXPORTED
+Section Refinements
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/Hab.var".
+
+alias id "m" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/m.var".
+
+alias id "n" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/n.var".
+
+alias id "P" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/P.var".
+
+alias id "Q" = "cic:/CoRN/ftc/Partitions/Definitions/Refinements/Q.var".
+
+(*#*
+We now define what it means for a partition [Q] to be a refinement of
+[P] and prove the main property of refinements.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Refinement.con".
+
+inline "cic:/CoRN/ftc/Partitions/Refinement_prop.con".
+
+(*#*
+We will also need to consider arbitrary sums %of the form
+\[\sum_{i=0}^{n-1}f(x_i)(a_{i+1}-a_i)\]%#of
+f(x<sub>i</sub>)(a<sub>i+1</sub>-a<sub>i</sub>)# where
+$x_i\in[a_i,a_{i+1}]$#x<sub>i</sub>∈[a<sub>i</sub>,a<sub>i+1</sub>]#.
+For this, we again need a choice function [x] which has to satisfy
+some condition. We define the condition and the sum for a fixed [P]:
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Points_in_Partition.con".
+
+inline "cic:/CoRN/ftc/Partitions/Pts_part_lemma.con".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_Sum.con".
+
+(* UNEXPORTED
+End Refinements
+*)
+
+(* UNEXPORTED
+Implicit Arguments Points_in_Partition [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Partition_Sum [a b Hab n P g F].
+*)
+
+(*#* **Constructions
+
+We now formalize some trivial and helpful constructions.
+
+%\begin{convention}% We will assume a fixed compact interval [[a,b]], denoted by [I].
+%\end{convention}%
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/Definitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/Definitions/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/Definitions/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Partitions/Definitions/I.con" "Definitions__".
+
+(* end hide *)
+
+(* UNEXPORTED
+Section Getting_Points
+*)
+
+(*#*
+From a partition we always have a canonical choice of points at which
+to evaluate a function: just take all but the last points of the
+partition.
+
+%\begin{convention}% Let [Q] be a partition of [I] with [m] points.
+%\end{convention}%
+*)
+
+alias id "m" = "cic:/CoRN/ftc/Partitions/Definitions/Getting_Points/m.var".
+
+alias id "Q" = "cic:/CoRN/ftc/Partitions/Definitions/Getting_Points/Q.var".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_imp_points.con".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_imp_points_1.con".
+
+inline "cic:/CoRN/ftc/Partitions/Partition_imp_points_2.con".
+
+(* UNEXPORTED
+End Getting_Points
+*)
+
+(*#*
+As a corollary, given any continuous function [F] and a natural number we have an immediate choice of a sum of [F] in [[a,b]].
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Partitions/Definitions/F.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Partitions/Definitions/contF.var".
+
+inline "cic:/CoRN/ftc/Partitions/Even_Partition_Sum.con".
+
+(* UNEXPORTED
+End Definitions
+*)
+
+(* UNEXPORTED
+Implicit Arguments Partition [a b].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Partition_Dom [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Mesh [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments AntiMesh [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Pts [a b Hab lng].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Part_Mesh_List [n a b Hab].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Points_in_Partition [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Partition_Sum [a b Hab n P g F].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Even_Partition [a b].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Even_Partition_Sum [a b].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Refinement [a b Hab n m].
+*)
+
+(* UNEXPORTED
+Hint Resolve start finish: algebra.
+*)
+
+(* UNEXPORTED
+Section Lemmas
+*)
+
+(*#* ** Properties of the mesh
+
+If a partition has more than one point then its mesh list is nonempty.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/length_Part_Mesh_List.con".
+
+(*#*
+Any element of the auxiliary list defined to calculate the mesh of a partition has a very specific form.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Part_Mesh_List_lemma.con".
+
+(*#*
+Mesh and antimesh are always nonnegative.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Mesh_nonneg.con".
+
+inline "cic:/CoRN/ftc/Partitions/AntiMesh_nonneg.con".
+
+(*#*
+Most important, [AntiMesh] and [Mesh] provide lower and upper bounds
+for the distance between any two consecutive points in a partition.
+
+%\begin{convention}% Let [I] be [[a,b]] and [P] be a partition of [I]
+with [n] points.
+%\end{convention}%
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/Lemmas/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/Lemmas/b.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Partitions/Lemmas/I.con" "Lemmas__".
+
+(* end hide *)
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/Lemmas/Hab.var".
+
+alias id "n" = "cic:/CoRN/ftc/Partitions/Lemmas/n.var".
+
+alias id "P" = "cic:/CoRN/ftc/Partitions/Lemmas/P.var".
+
+inline "cic:/CoRN/ftc/Partitions/Mesh_lemma.con".
+
+inline "cic:/CoRN/ftc/Partitions/AntiMesh_lemma.con".
+
+inline "cic:/CoRN/ftc/Partitions/Mesh_leEq.con".
+
+(* UNEXPORTED
+End Lemmas
+*)
+
+(* UNEXPORTED
+Section Even_Partitions
+*)
+
+(*#* More technical stuff. Two equal partitions have the same mesh.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Mesh_wd.con".
+
+inline "cic:/CoRN/ftc/Partitions/Mesh_wd'.con".
+
+(*#*
+The mesh of an even partition is easily calculated.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/even_partition_Mesh.con".
+
+(*#* ** Miscellaneous
+%\begin{convention}% Throughout this section, let [a,b:IR] and [I] be [[a,b]].
+%\end{convention}%
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/Even_Partitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/Even_Partitions/b.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Partitions/Even_Partitions/I.con" "Even_Partitions__".
+
+(* end hide *)
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/Even_Partitions/Hab.var".
+
+(*#*
+An interesting property: in a partition, if [ai [<] aj] then [i < j].
+*)
+
+inline "cic:/CoRN/ftc/Partitions/Partition_Points_mon.con".
+
+inline "cic:/CoRN/ftc/Partitions/refinement_resp_mult.con".
+
+(*#*
+%\begin{convention}% Assume [m,n] are positive natural numbers and
+denote by [P] and [Q] the even partitions with, respectively, [m] and
+[n] points.
+%\end{convention}%
+
+Even partitions always have a common refinement.
+*)
+
+alias id "m" = "cic:/CoRN/ftc/Partitions/Even_Partitions/m.var".
+
+alias id "n" = "cic:/CoRN/ftc/Partitions/Even_Partitions/n.var".
+
+alias id "Hm" = "cic:/CoRN/ftc/Partitions/Even_Partitions/Hm.var".
+
+alias id "Hn" = "cic:/CoRN/ftc/Partitions/Even_Partitions/Hn.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Partitions/Even_Partitions/P.con" "Even_Partitions__".
+
+inline "cic:/CoRN/ftc/Partitions/Even_Partitions/Q.con" "Even_Partitions__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Partitions/even_partition_refinement.con".
+
+(* UNEXPORTED
+End Even_Partitions
+*)
+
+(* UNEXPORTED
+Section More_Definitions
+*)
+
+(*#* ** Separation
+
+Some auxiliary definitions. A partition is said to be separated if all its points are distinct.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/More_Definitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/More_Definitions/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/More_Definitions/Hab.var".
+
+inline "cic:/CoRN/ftc/Partitions/_Separated.con".
+
+(*#*
+Two partitions are said to be (mutually) separated if they are both
+separated and all their points are distinct (except for the
+endpoints).
+
+%\begin{convention}% Let [P,Q] be partitions of [I] with,
+respectively, [n] and [m] points.
+%\end{convention}%
+*)
+
+alias id "n" = "cic:/CoRN/ftc/Partitions/More_Definitions/n.var".
+
+alias id "m" = "cic:/CoRN/ftc/Partitions/More_Definitions/m.var".
+
+alias id "P" = "cic:/CoRN/ftc/Partitions/More_Definitions/P.var".
+
+alias id "Q" = "cic:/CoRN/ftc/Partitions/More_Definitions/Q.var".
+
+inline "cic:/CoRN/ftc/Partitions/Separated.con".
+
+(* UNEXPORTED
+End More_Definitions
+*)
+
+(* UNEXPORTED
+Implicit Arguments _Separated [a b Hab n].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Separated [a b Hab n m].
+*)
+
+(* UNEXPORTED
+Section Sep_Partitions
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Partitions/Sep_Partitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Partitions/Sep_Partitions/b.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Partitions/Sep_Partitions/I.con" "Sep_Partitions__".
+
+(* end hide *)
+
+alias id "Hab" = "cic:/CoRN/ftc/Partitions/Sep_Partitions/Hab.var".
+
+(*#*
+The antimesh of a separated partition is always positive.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/pos_AntiMesh.con".
+
+(*#*
+A partition can have only one point iff the endpoints of the interval
+are the same; moreover, if the partition is separated and the
+endpoints of the interval are the same then it must have one point.
+*)
+
+inline "cic:/CoRN/ftc/Partitions/partition_length_zero.con".
+
+inline "cic:/CoRN/ftc/Partitions/_Separated_imp_length_zero.con".
+
+inline "cic:/CoRN/ftc/Partitions/partition_less_imp_gt_zero.con".
+
+(* UNEXPORTED
+End Sep_Partitions
+*)
+
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