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19 (* $Id: Derivative.v,v 1.7 2004/04/23 10:00:58 lcf Exp $ *)
21 include "ftc/Continuity.ma".
29 We will now proceed toward the development of differential calculus.
30 To begin with, the main notion is that of derivative.
32 At this stage we will not define a notion of differentiable function,
33 mainly because the natural definition (that of being a function which
34 has some derivative) poses some technical problems; thus, we will
35 postpone that part of our work to a subsequent stage.
37 Derivative is a binary relation in the type of partial functions,
38 dependent (once again) on a compact interval with distinct
39 endpoints#. #%\footnote{%As before, we do not define pointwise
40 differentiability, mainly for coherence reasons. See Bishop [1967]
41 for a discussion on the relative little interest of that concept.%}.%
42 The reason for requiring the endpoints to be apart is mainly to be
43 able to derive the usual properties of the derivative
44 relation---namely, that any two derivatives of the same function must
47 %\begin{convention}% Let [a,b:IR] with [a [<] b] and denote by [I] the
48 interval [[a,b]]. Throughout this chapter, [F, F', G, G'] and [H]
49 will be partial functions with domains respectively [P, P', Q, Q'] and
54 alias id "a" = "cic:/CoRN/ftc/Derivative/Definitions/a.var".
56 alias id "b" = "cic:/CoRN/ftc/Derivative/Definitions/b.var".
58 alias id "Hab'" = "cic:/CoRN/ftc/Derivative/Definitions/Hab'.var".
62 inline procedural "cic:/CoRN/ftc/Derivative/Definitions/Hab.con" "Definitions__" as definition.
64 inline procedural "cic:/CoRN/ftc/Derivative/Definitions/I.con" "Definitions__" as definition.
68 alias id "F" = "cic:/CoRN/ftc/Derivative/Definitions/F.var".
72 inline procedural "cic:/CoRN/ftc/Derivative/Definitions/P.con" "Definitions__" as definition.
76 inline procedural "cic:/CoRN/ftc/Derivative/Derivative_I.con" as definition.
83 Implicit Arguments Derivative_I [a b].
87 Section Basic_Properties
90 (*#* **Basic Properties
93 alias id "a" = "cic:/CoRN/ftc/Derivative/Basic_Properties/a.var".
95 alias id "b" = "cic:/CoRN/ftc/Derivative/Basic_Properties/b.var".
97 alias id "Hab'" = "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab'.var".
101 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab.con" "Basic_Properties__" as definition.
103 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/I.con" "Basic_Properties__" as definition.
108 Like we did for equality, we begin by stating a lemma that makes proofs of derivation easier in practice.
111 inline procedural "cic:/CoRN/ftc/Derivative/Derivative_I_char.con" as lemma.
116 Derivative is a well defined relation; we will make this explicit for both arguments:
119 alias id "F" = "cic:/CoRN/ftc/Derivative/Basic_Properties/F.var".
121 alias id "G" = "cic:/CoRN/ftc/Derivative/Basic_Properties/G.var".
123 alias id "H" = "cic:/CoRN/ftc/Derivative/Basic_Properties/H.var".
127 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/P.con" "Basic_Properties__" as definition.
129 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/Q.con" "Basic_Properties__" as definition.
131 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/R.con" "Basic_Properties__" as definition.
135 inline procedural "cic:/CoRN/ftc/Derivative/Derivative_I_wdl.con" as lemma.
137 inline procedural "cic:/CoRN/ftc/Derivative/Derivative_I_wdr.con" as lemma.
141 inline procedural "cic:/CoRN/ftc/Derivative/Basic_Properties/Derivative_I_unique_lemma.con" "Basic_Properties__" as definition.
146 Derivative is unique.
149 inline procedural "cic:/CoRN/ftc/Derivative/Derivative_I_unique.con" as lemma.
152 Finally, the set where we are considering the relation is included in the domain of both functions.
155 inline procedural "cic:/CoRN/ftc/Derivative/derivative_imp_inc.con" as lemma.
157 inline procedural "cic:/CoRN/ftc/Derivative/derivative_imp_inc'.con" as lemma.
160 Any function that is or has a derivative is continuous.
163 alias id "Hab''" = "cic:/CoRN/ftc/Derivative/Basic_Properties/Hab''.var".
165 inline procedural "cic:/CoRN/ftc/Derivative/deriv_imp_contin'_I.con" as lemma.
167 inline procedural "cic:/CoRN/ftc/Derivative/deriv_imp_contin_I.con" as lemma.
174 If [G] is the derivative of [F] in a given interval, then [G] is also the derivative of [F] in any smaller interval.
177 inline procedural "cic:/CoRN/ftc/Derivative/included_imp_deriv.con" as lemma.