--- /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/Continuity".
+
+include "CoRN.ma".
+
+(* $Id: Continuity.v,v 1.6 2004/04/23 10:00:58 lcf Exp $ *)
+
+(*#* printing Norm_Funct %\ensuremath{\|\cdot\|}% *)
+
+include "reals/NRootIR.ma".
+
+include "ftc/FunctSums.ma".
+
+(* UNEXPORTED
+Section Definitions_and_Basic_Results
+*)
+
+(*#* *Continuity
+
+Constructively, continuity is the most fundamental property of any
+function---so strongly that no example is known of a constructive
+function that is not continuous. However, the classical definition of
+continuity is not good for our purposes, as it is not true, for
+example, that a function which is continuous in a compact interval is
+uniformly continuous in that same interval (for a discussion of this
+see Bishop 1967). Thus, our notion of continuity will be the uniform
+one#. #%\footnote{%Similar remarks apply to convergence of sequences
+of functions, which we will define ahead, and elsewhere; we will
+refrain from discussing this issue at those places.%}.%
+
+%\begin{convention}% Throughout this section, [a] and [b] will be real
+numbers, [I] will denote the compact interval [[a,b]] and
+[F, G, H] will denote arbitrary partial functions with domains
+respectively [P, Q] and [R].
+%\end{convention}%
+
+** Definitions and Basic Results
+
+Here we define continuity and prove some basic properties of continuous functions.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/I.con" "Definitions_and_Basic_Results__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/F.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/P.con" "Definitions_and_Basic_Results__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I.con".
+
+(*#*
+For convenience, we distinguish the two properties of continuous functions.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/contin_imp_inc.con".
+
+inline "cic:/CoRN/ftc/Continuity/contin_prop.con".
+
+(*#*
+Assume [F] to be continuous in [I]. Then it has a least upper bound and a greater lower bound on [I].
+*)
+
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/contF.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hinc'.con" "Definitions_and_Basic_Results__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_tb_image.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_lub.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_glb.con".
+
+(*#*
+We now make this glb and lub into operators.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/lub_funct.con".
+
+inline "cic:/CoRN/ftc/Continuity/glb_funct.con".
+
+(*#*
+These operators have the expected properties.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/lub_is_lub.con".
+
+inline "cic:/CoRN/ftc/Continuity/glb_is_glb.con".
+
+inline "cic:/CoRN/ftc/Continuity/glb_prop.con".
+
+inline "cic:/CoRN/ftc/Continuity/lub_prop.con".
+
+(*#*
+The norm of a function is defined as being the supremum of its absolute value; that is equivalent to the following definition (which is often more convenient to use).
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Norm_Funct.con".
+
+(*#*
+The norm effectively bounds the absolute value of a function.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/norm_bnd_AbsIR.con".
+
+(*#*
+The following is another way of characterizing the norm:
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_imp_abs_lub.con".
+
+(*#*
+We now prove some basic properties of the norm---namely that it is positive, and that it provides a least upper bound for the absolute value of its argument.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/positive_norm.con".
+
+inline "cic:/CoRN/ftc/Continuity/norm_fun_lub.con".
+
+inline "cic:/CoRN/ftc/Continuity/leEq_Norm_Funct.con".
+
+inline "cic:/CoRN/ftc/Continuity/less_Norm_Funct.con".
+
+(* UNEXPORTED
+End Definitions_and_Basic_Results
+*)
+
+(* UNEXPORTED
+Implicit Arguments Continuous_I [a b].
+*)
+
+(* UNEXPORTED
+Implicit Arguments Norm_Funct [a b Hab F].
+*)
+
+(* UNEXPORTED
+Section Local_Results
+*)
+
+(*#* **Algebraic Properties
+
+We now state and prove some results about continuous functions. Assume that [I] is included in the domain of both [F] and [G].
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Continuity/Local_Results/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Continuity/Local_Results/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Local_Results/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Local_Results/I.con" "Local_Results__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Continuity/Local_Results/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Continuity/Local_Results/G.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Local_Results/P.con" "Local_Results__".
+
+inline "cic:/CoRN/ftc/Continuity/Local_Results/Q.con" "Local_Results__".
+
+(* end hide *)
+
+alias id "incF" = "cic:/CoRN/ftc/Continuity/Local_Results/incF.var".
+
+alias id "incG" = "cic:/CoRN/ftc/Continuity/Local_Results/incG.var".
+
+(*#*
+The first result does not require the function to be continuous; however, its preconditions are easily verified by continuous functions, which justifies its inclusion in this section.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change.con".
+
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change_pos.con".
+
+inline "cic:/CoRN/ftc/Continuity/cont_no_sign_change_neg.con".
+
+(*#*
+Being continuous is an extensional property.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_wd.con".
+
+(*#*
+A continuous function remains continuous if you restrict its domain.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/included_imp_contin.con".
+
+(*#*
+Constant functions and identity are continuous.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_const.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_id.con".
+
+(*#*
+Assume [F] and [G] are continuous in [I]. Then functions derived from these through algebraic operations are also continuous, provided (in the case of reciprocal and division) some extra conditions are met.
+*)
+
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Local_Results/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Local_Results/contG.var".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_plus.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_inv.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_mult.con".
+
+(* UNEXPORTED
+Opaque AbsIR Max.
+*)
+
+(* UNEXPORTED
+Transparent AbsIR Max.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_max.con".
+
+(* begin show *)
+
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg'.var".
+
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg''.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_recip.con".
+
+(* UNEXPORTED
+End Local_Results
+*)
+
+(* UNEXPORTED
+Hint Resolve contin_imp_inc: included.
+*)
+
+(* UNEXPORTED
+Section Corolaries
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Continuity/Corolaries/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Continuity/Corolaries/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Corolaries/Hab.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Corolaries/I.con" "Corolaries__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Continuity/Corolaries/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Continuity/Corolaries/G.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Corolaries/P.con" "Corolaries__".
+
+inline "cic:/CoRN/ftc/Continuity/Corolaries/Q.con" "Corolaries__".
+
+(* end hide *)
+
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Corolaries/contF.var".
+
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Corolaries/contG.var".
+
+(*#*
+The corresponding properties for subtraction, division and
+multiplication by a scalar are easily proved as corollaries;
+exponentiation is proved by induction, appealing to the results on
+product and constant functions.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_minus.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_scal.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_nth.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_min.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_abs.con".
+
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg'.var".
+
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg''.var".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_div.con".
+
+(* UNEXPORTED
+End Corolaries
+*)
+
+(* UNEXPORTED
+Section Other
+*)
+
+(* UNEXPORTED
+Section Sums
+*)
+
+(*#*
+We finally prove that the sum of an arbitrary family of continuous functions is still a continuous function.
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Continuity/Other/Sums/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Continuity/Other/Sums/b.var".
+
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Other/Sums/I.con" "Other__Sums__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sum0.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sumx.con".
+
+inline "cic:/CoRN/ftc/Continuity/Continuous_I_Sum.con".
+
+(* UNEXPORTED
+End Sums
+*)
+
+(*#*
+For practical purposes, these characterization results are useful:
+*)
+
+inline "cic:/CoRN/ftc/Continuity/lub_charact.con".
+
+inline "cic:/CoRN/ftc/Continuity/glb_charact.con".
+
+(*#*
+The following result is also extremely useful, as it allows us to set a lower bound on the glb of a function.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/leEq_glb.con".
+
+(*#*
+The norm is also an extensional property.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/Norm_Funct_wd.con".
+
+(*#*
+The value of the norm is covariant with the length of the interval.
+*)
+
+inline "cic:/CoRN/ftc/Continuity/included_imp_norm_leEq.con".
+
+(* UNEXPORTED
+End Other
+*)
+
+(* UNEXPORTED
+Hint Resolve Continuous_I_const Continuous_I_id Continuous_I_plus
+ Continuous_I_inv Continuous_I_minus Continuous_I_mult Continuous_I_scal
+ Continuous_I_recip Continuous_I_max Continuous_I_min Continuous_I_div
+ Continuous_I_nth Continuous_I_abs: continuous.
+*)
+
projects / helm.git / blobdiff