--- /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/Integral".
+
+include "CoRN.ma".
+
+(* $Id: Integral.v,v 1.10 2004/04/23 10:00:59 lcf Exp $ *)
+
+include "ftc/RefLemma.ma".
+
+(*#* printing integral %\ensuremath{\int_I}% #∫<sub>I</sub># *)
+
+(*#* printing Integral %\ensuremath{\int_I}% #∫<sub>I</sub># *)
+
+(* begin hide *)
+
+(* UNEXPORTED
+Section Lemmas
+*)
+
+inline "cic:/CoRN/ftc/Integral/Lemmas/Sumx_wd_weird.con" "Lemmas__".
+
+inline "cic:/CoRN/ftc/Integral/Sumx_weird_lemma.con".
+
+(* UNEXPORTED
+End Lemmas
+*)
+
+(* end hide *)
+
+(* UNEXPORTED
+Section Integral
+*)
+
+(*#* *Integral
+
+Having proved the main properties of partitions and refinements, we
+will define the integral of a continuous function [F] in the interval
+[[a,b]] as the limit of the sequence of Sums of $F$ for even
+partitions of increasing number of points.
+
+%\begin{convention}% All throughout, [a,b] will be real numbers, the
+interval [[a,b]] will be denoted by [I] and [F,G] will be
+continuous functions in [I].
+%\end{convention}%
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Integral/Integral/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Integral/Integral/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Integral/Integral/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/Integral/I.con" "Integral__".
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Integral/F.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Integral/Integral/contF.var".
+
+inline "cic:/CoRN/ftc/Integral/Integral/contF'.con" "Integral__".
+
+(* end hide *)
+
+(* UNEXPORTED
+Section Darboux_Sum
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_seq.con".
+
+inline "cic:/CoRN/ftc/Integral/Cauchy_Darboux_Seq.con".
+
+inline "cic:/CoRN/ftc/Integral/integral.con".
+
+(* UNEXPORTED
+End Darboux_Sum
+*)
+
+(* UNEXPORTED
+Section Integral_Thm
+*)
+
+(*#*
+The following shows that in fact the integral of [F] is the limit of
+any sequence of partitions of mesh converging to 0.
+
+%\begin{convention}% Let [e] be a positive real number and [P] be a
+partition of [I] with [n] points and mesh smaller than the
+modulus of continuity of [F] for [e]. Let [fP] be a choice of points
+respecting [P].
+%\end{convention}%
+*)
+
+alias id "n" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/n.var".
+
+alias id "P" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/P.var".
+
+alias id "e" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/e.var".
+
+alias id "He" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/He.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/d.con" "Integral__Integral_Thm__".
+
+(* end hide *)
+
+alias id "HmeshP" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/HmeshP.var".
+
+alias id "fP" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/fP.var".
+
+alias id "HfP" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/HfP.var".
+
+alias id "HfP'" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/HfP'.var".
+
+alias id "incF" = "cic:/CoRN/ftc/Integral/Integral/Integral_Thm/incF.var".
+
+inline "cic:/CoRN/ftc/Integral/partition_Sum_conv_integral.con".
+
+(* UNEXPORTED
+End Integral_Thm
+*)
+
+(* UNEXPORTED
+End Integral
+*)
+
+(* UNEXPORTED
+Section Basic_Properties
+*)
+
+(*#*
+The usual extensionality and strong extensionality results hold.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Integral/Basic_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Integral/Basic_Properties/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Integral/Basic_Properties/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/Basic_Properties/I.con" "Basic_Properties__".
+
+(* end hide *)
+
+(* NOTATION
+Notation Integral := (integral _ _ Hab).
+*)
+
+(* UNEXPORTED
+Section Well_Definedness
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Basic_Properties/Well_Definedness/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Integral/Basic_Properties/Well_Definedness/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Integral/Basic_Properties/Well_Definedness/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Integral/Basic_Properties/Well_Definedness/contG.var".
+
+inline "cic:/CoRN/ftc/Integral/integral_strext.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_strext'.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_wd.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_wd'.con".
+
+(* UNEXPORTED
+End Well_Definedness
+*)
+
+(* UNEXPORTED
+Section Linearity_and_Monotonicity
+*)
+
+(* UNEXPORTED
+Opaque Even_Partition.
+*)
+
+(*#*
+The integral is a linear and monotonous function; in order to prove these facts we also need the important equalities $\int_a^bdx=b-a$#∫<sub>a</sub><sup>b</sup>dx=b-a# and $\int_a^af(x)dx=0$#∫<sub>a</sub><sup>a</sup>=0#.
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_one.con".
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity/contG.var".
+
+inline "cic:/CoRN/ftc/Integral/integral_comm_scal.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_plus.con".
+
+(* UNEXPORTED
+Transparent Even_Partition.
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_empty.con".
+
+(* UNEXPORTED
+End Linearity_and_Monotonicity
+*)
+
+(* UNEXPORTED
+Section Linearity_and_Monotonicity'
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/contG.var".
+
+(*#*
+%\begin{convention}% Let [alpha, beta : IR] and assume that
+[h := alpha{**}F{+}beta{**}G] is continuous.
+%\end{convention}%
+*)
+
+alias id "alpha" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/alpha.var".
+
+alias id "beta" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/beta.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/h.con" "Basic_Properties__Linearity_and_Monotonicity'__".
+
+(* end hide *)
+
+alias id "contH" = "cic:/CoRN/ftc/Integral/Basic_Properties/Linearity_and_Monotonicity'/contH.var".
+
+inline "cic:/CoRN/ftc/Integral/linear_integral.con".
+
+(* UNEXPORTED
+Opaque nring.
+*)
+
+inline "cic:/CoRN/ftc/Integral/monotonous_integral.con".
+
+(* UNEXPORTED
+Transparent nring.
+*)
+
+(* UNEXPORTED
+End Linearity_and_Monotonicity'
+*)
+
+(* UNEXPORTED
+Section Corollaries
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Basic_Properties/Corollaries/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Integral/Basic_Properties/Corollaries/G.var".
+
+alias id "contF" = "cic:/CoRN/ftc/Integral/Basic_Properties/Corollaries/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Integral/Basic_Properties/Corollaries/contG.var".
+
+(*#*
+As corollaries we can calculate integrals of group operations applied to functions.
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_const.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_minus.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_inv.con".
+
+(*#*
+We can also bound integrals by bounding the integrated functions.
+*)
+
+inline "cic:/CoRN/ftc/Integral/lb_integral.con".
+
+inline "cic:/CoRN/ftc/Integral/ub_integral.con".
+
+inline "cic:/CoRN/ftc/Integral/integral_leEq_norm.con".
+
+(* UNEXPORTED
+End Corollaries
+*)
+
+(* UNEXPORTED
+Section Integral_Sum
+*)
+
+(*#*
+We now relate the sum of integrals in adjoining intervals to the
+integral over the union of those intervals.
+
+%\begin{convention}% Let [c] be a real number such that
+$c\in[a,b]$#c∈[a,b]#.
+%\end{convention}%
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/F.var".
+
+alias id "c" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/c.var".
+
+alias id "Hac" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Hac.var".
+
+alias id "Hcb" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Hcb.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Hab'.var".
+
+alias id "Hac'" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Hac'.var".
+
+alias id "Hcb'" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Hcb'.var".
+
+(* UNEXPORTED
+Section Partition_Join
+*)
+
+(*#*
+We first prove that every pair of partitions, one of [[a,c]]
+and another of [[c,b]] defines a partition of [[a,b]] the mesh
+of which is less or equal to the maximum of the mesh of the original
+partitions (actually it is equal, but we don't need the other
+inequality).
+
+%\begin{convention}% Let [P,Q] be partitions respectively of
+[[a,c]] and [[c,b]] with [n] and [m] points.
+%\end{convention}%
+*)
+
+alias id "n" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/n.var".
+
+alias id "m" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/m.var".
+
+alias id "P" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/P.var".
+
+alias id "Q" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/Q.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_aux.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_fun.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/pjf_1.con".
+
+inline "cic:/CoRN/ftc/Integral/pjf_2.con".
+
+inline "cic:/CoRN/ftc/Integral/pjf_2'.con".
+
+inline "cic:/CoRN/ftc/Integral/pjf_3.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_prf1.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_prf2.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_start.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_finish.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join.con".
+
+(* end hide *)
+
+(*#*
+%\begin{convention}% [fP, fQ] are choices of points respecting [P] and [Q].
+%\end{convention}%
+*)
+
+alias id "fP" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/fP.var".
+
+alias id "HfP" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/HfP.var".
+
+alias id "HfP'" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/HfP'.var".
+
+alias id "fQ" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/fQ.var".
+
+alias id "HfQ" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/HfQ.var".
+
+alias id "HfQ'" = "cic:/CoRN/ftc/Integral/Basic_Properties/Integral_Sum/Partition_Join/HfQ'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_aux'.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_pts.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Integral/pjp_1.con".
+
+inline "cic:/CoRN/ftc/Integral/pjp_2.con".
+
+inline "cic:/CoRN/ftc/Integral/pjp_3.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_Pts_in_partition.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_Pts_wd.con".
+
+(* UNEXPORTED
+Opaque partition_join.
+*)
+
+(* UNEXPORTED
+Transparent partition_join.
+*)
+
+(* UNEXPORTED
+Opaque minus.
+*)
+
+(* UNEXPORTED
+Transparent minus.
+*)
+
+(* UNEXPORTED
+Opaque minus.
+*)
+
+(* UNEXPORTED
+Transparent minus.
+*)
+
+inline "cic:/CoRN/ftc/Integral/partition_join_Sum_lemma.con".
+
+inline "cic:/CoRN/ftc/Integral/partition_join_mesh.con".
+
+(* UNEXPORTED
+End Partition_Join
+*)
+
+(*#*
+With these results in mind, the following is a trivial consequence:
+*)
+
+(* UNEXPORTED
+Opaque Even_Partition.
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_plus_integral.con".
+
+(* UNEXPORTED
+End Integral_Sum
+*)
+
+(* UNEXPORTED
+Transparent Even_Partition.
+*)
+
+(* UNEXPORTED
+End Basic_Properties
+*)
+
+(*#*
+The following are simple consequences of this result and of previous ones.
+*)
+
+inline "cic:/CoRN/ftc/Integral/integral_less_norm.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Integral/integral_gt_zero.con".
+
+(* end hide *)
+
+(*#* remove printing Integral *)
+
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