branch for universe

[helm.git] / matita / contribs / CoRN-Decl / ftc / Derivative.ma

diff --git a/matita/contribs/CoRN-Decl/ftc/Derivative.ma b/matita/contribs/CoRN-Decl/ftc/Derivative.ma
new file mode 100644 (file)
index 0000000..8c983b1
--- /dev/null
+++ b/matita/contribs/CoRN-Decl/ftc/Derivative.ma
@@ -0,0 +1,180 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/Derivative".
+
+include "CoRN.ma".
+
+(* $Id: Derivative.v,v 1.7 2004/04/23 10:00:58 lcf Exp $ *)
+
+include "ftc/Continuity.ma".
+
+(* UNEXPORTED
+Section Definitions
+*)
+
+(*#* *Derivatives
+
+We will now proceed toward the development of differential calculus.
+To begin with, the main notion is that of derivative.
+
+At this stage we will not define a notion of differentiable function,
+mainly because the natural definition (that of being a function which
+has some derivative) poses some technical problems; thus, we will
+postpone that part of our work to a subsequent stage.
+
+Derivative is a binary relation in the type of partial functions,
+dependent (once again) on a compact interval with distinct
+endpoints#. #%\footnote{%As before, we do not define pointwise
+differentiability, mainly for coherence reasons.  See Bishop [1967]
+for a discussion on the relative little interest of that concept.%}.%
+The reason for requiring the endpoints to be apart is mainly to be
+able to derive the usual properties of the derivative
+relation---namely, that any two derivatives of the same function must
+coincide.
+
+%\begin{convention}% Let [a,b:IR] with [a [<] b] and denote by [I] the
+interval [[a,b]].  Throughout this chapter, [F, F', G, G'] and [H]
+will be partial functions with domains respectively [P, P', Q, Q'] and
+[R].
+%\end{convention}%
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Derivative/Definitions/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Derivative/Definitions/b.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Derivative/Definitions/Hab'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Definitions/Hab.con" "Definitions__".
+
+inline "cic:/CoRN/ftc/Derivative/Definitions/I.con" "Definitions__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Derivative/Definitions/F.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Definitions/P.con" "Definitions__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Derivative_I.con".
+
+(* UNEXPORTED
+End Definitions
+*)
+
+(* UNEXPORTED
+Implicit Arguments Derivative_I [a b].
+*)
+
+(* UNEXPORTED
+Section Basic_Properties
+*)
+
+(*#* **Basic Properties
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Derivative/Basic_Properties/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Derivative/Basic_Properties/b.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab.con" "Basic_Properties__".
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/I.con" "Basic_Properties__".
+
+(* end hide *)
+
+(*#*
+Like we did for equality, we begin by stating a lemma that makes proofs of derivation easier in practice.
+*)
+
+inline "cic:/CoRN/ftc/Derivative/Derivative_I_char.con".
+
+(* end hide *)
+
+(*#*
+Derivative is a well defined relation; we will make this explicit for both arguments:
+*)
+
+alias id "F" = "cic:/CoRN/ftc/Derivative/Basic_Properties/F.var".
+
+alias id "G" = "cic:/CoRN/ftc/Derivative/Basic_Properties/G.var".
+
+alias id "H" = "cic:/CoRN/ftc/Derivative/Basic_Properties/H.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/P.con" "Basic_Properties__".
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/Q.con" "Basic_Properties__".
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/R.con" "Basic_Properties__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Derivative_I_wdl.con".
+
+inline "cic:/CoRN/ftc/Derivative/Derivative_I_wdr.con".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Derivative/Basic_Properties/Derivative_I_unique_lemma.con" "Basic_Properties__".
+
+(* end hide *)
+
+(*#*
+Derivative is unique.
+*)
+
+inline "cic:/CoRN/ftc/Derivative/Derivative_I_unique.con".
+
+(*#*
+Finally, the set where we are considering the relation is included in the domain of both functions.
+*)
+
+inline "cic:/CoRN/ftc/Derivative/derivative_imp_inc.con".
+
+inline "cic:/CoRN/ftc/Derivative/derivative_imp_inc'.con".
+
+(*#*
+Any function that is or has a derivative is continuous.
+*)
+
+alias id "Hab''" = "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab''.var".
+
+inline "cic:/CoRN/ftc/Derivative/deriv_imp_contin'_I.con".
+
+inline "cic:/CoRN/ftc/Derivative/deriv_imp_contin_I.con".
+
+(* UNEXPORTED
+End Basic_Properties
+*)
+
+(*#*
+If [G] is the derivative of [F] in a given interval, then [G] is also the derivative of [F] in any smaller interval.
+*)
+
+inline "cic:/CoRN/ftc/Derivative/included_imp_deriv.con".
+