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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: Rolle.v,v 1.6 2004/04/23 10:01:01 lcf Exp $ *)
21 include "tactics/DiffTactics2.ma".
23 include "ftc/MoreFunctions.ma".
31 We now begin to work with partial functions. We begin by stating and proving Rolle's theorem in various forms and some of its corollaries.
33 %\begin{convention}% Assume that:
34 - [a,b:IR] with [a [<] b] and denote by [I] the interval [[a,b]];
35 - [F,F'] are partial functions such that [F'] is the derivative of [F] in [I];
36 - [e] is a positive real number.
44 cic:/CoRN/ftc/Rolle/Rolle/a.var
48 cic:/CoRN/ftc/Rolle/Rolle/b.var
52 cic:/CoRN/ftc/Rolle/Rolle/Hab'.var
55 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hab.con" "Rolle__" as definition.
57 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/I.con" "Rolle__" as definition.
60 cic:/CoRN/ftc/Rolle/Rolle/F.var
64 cic:/CoRN/ftc/Rolle/Rolle/F'.var
68 cic:/CoRN/ftc/Rolle/Rolle/derF.var
72 cic:/CoRN/ftc/Rolle/Rolle/Ha.var
76 cic:/CoRN/ftc/Rolle/Rolle/Hb.var
84 cic:/CoRN/ftc/Rolle/Rolle/Fab.var
92 cic:/CoRN/ftc/Rolle/Rolle/e.var
96 cic:/CoRN/ftc/Rolle/Rolle/He.var
99 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/contF'.con" "Rolle__" as definition.
101 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/derivF.con" "Rolle__" as definition.
103 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma2.con" "Rolle__" as definition.
105 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/df.con" "Rolle__" as definition.
107 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hdf.con" "Rolle__" as definition.
109 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hf.con" "Rolle__" as definition.
111 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma3.con" "Rolle__" as definition.
113 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/df'.con" "Rolle__" as definition.
115 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hdf'.con" "Rolle__" as definition.
117 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hf'.con" "Rolle__" as definition.
119 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/d.con" "Rolle__" as definition.
121 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hd.con" "Rolle__" as definition.
123 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/incF.con" "Rolle__" as definition.
125 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/n.con" "Rolle__" as definition.
127 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/fcp.con" "Rolle__" as definition.
129 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma1.con" "Rolle__" as definition.
131 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/incF'.con" "Rolle__" as definition.
133 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/fcp'.con" "Rolle__" as definition.
136 Notation cp := (compact_part a b Hab' d Hd).
139 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma4.con" "Rolle__" as definition.
141 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma5.con" "Rolle__" as definition.
143 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma6.con" "Rolle__" as definition.
145 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma7.con" "Rolle__" as definition.
147 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/j.con" "Rolle__" as definition.
149 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hj.con" "Rolle__" as definition.
151 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hj'.con" "Rolle__" as definition.
153 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/k.con" "Rolle__" as definition.
155 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hk.con" "Rolle__" as definition.
157 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Hk'.con" "Rolle__" as definition.
159 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma8.con" "Rolle__" as definition.
161 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma9.con" "Rolle__" as definition.
163 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma10.con" "Rolle__" as definition.
165 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma11.con" "Rolle__" as definition.
167 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma12.con" "Rolle__" as definition.
169 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma13.con" "Rolle__" as definition.
171 inline procedural "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma15.con" "Rolle__" as definition.
175 inline procedural "cic:/CoRN/ftc/Rolle/Rolle.con" as theorem.
182 Section Law_of_the_Mean
186 The following is a simple corollary:
190 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/a.var
194 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/b.var
198 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/Hab'.var
203 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/Hab.con" "Law_of_the_Mean__" as definition.
205 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/I.con" "Law_of_the_Mean__" as definition.
210 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/F.var
214 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/F'.var
218 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HF.var
224 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HA.var
228 cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HB.var
233 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean_I.con" as lemma.
244 We can also state these theorems without expliciting the derivative of [F].
248 cic:/CoRN/ftc/Rolle/Corollaries/a.var
252 cic:/CoRN/ftc/Rolle/Corollaries/b.var
256 cic:/CoRN/ftc/Rolle/Corollaries/Hab'.var
261 inline procedural "cic:/CoRN/ftc/Rolle/Corollaries/Hab.con" "Corollaries__" as definition.
266 cic:/CoRN/ftc/Rolle/Corollaries/F.var
272 cic:/CoRN/ftc/Rolle/Corollaries/HF.var
277 inline procedural "cic:/CoRN/ftc/Rolle/Rolle'.con" as theorem.
279 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean'_I.con" as lemma.
286 Section Generalizations
290 The mean law is more useful if we abstract [a] and [b] from the
291 context---allowing them in particular to be equal. In the case where
292 [F(a) [=] F(b)] we get Rolle's theorem again, so there is no need to
293 state it also in this form.
295 %\begin{convention}% Assume [I] is a proper interval, [F,F':PartIR].
300 cic:/CoRN/ftc/Rolle/Generalizations/I.var
304 cic:/CoRN/ftc/Rolle/Generalizations/pI.var
308 cic:/CoRN/ftc/Rolle/Generalizations/F.var
312 cic:/CoRN/ftc/Rolle/Generalizations/F'.var
318 cic:/CoRN/ftc/Rolle/Generalizations/derF.var
325 inline procedural "cic:/CoRN/ftc/Rolle/Generalizations/incF.con" "Generalizations__" as definition.
327 inline procedural "cic:/CoRN/ftc/Rolle/Generalizations/incF'.con" "Generalizations__" as definition.
331 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean.con" as theorem.
334 We further generalize the mean law by writing as an explicit bound.
337 inline procedural "cic:/CoRN/ftc/Rolle/Law_of_the_Mean_ineq.con" as theorem.