--- /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/Rolle".
+
+include "CoRN.ma".
+
+(* $Id: Rolle.v,v 1.6 2004/04/23 10:01:01 lcf Exp $ *)
+
+include "tactics/DiffTactics2.ma".
+
+include "ftc/MoreFunctions.ma".
+
+(* UNEXPORTED
+Section Rolle
+*)
+
+(*#* *Rolle's Theorem
+
+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.
+
+%\begin{convention}% Assume that:
+ - [a,b:IR] with [a [<] b] and denote by [I] the interval [[a,b]];
+ - [F,F'] are partial functions such that [F'] is the derivative of [F] in [I];
+ - [e] is a positive real number.
+
+%\end{convention}%
+*)
+
+(* begin hide *)
+
+alias id "a" = "cic:/CoRN/ftc/Rolle/Rolle/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Rolle/Rolle/b.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Rolle/Rolle/Hab'.var".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hab.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/I.con" "Rolle__".
+
+alias id "F" = "cic:/CoRN/ftc/Rolle/Rolle/F.var".
+
+alias id "F'" = "cic:/CoRN/ftc/Rolle/Rolle/F'.var".
+
+alias id "derF" = "cic:/CoRN/ftc/Rolle/Rolle/derF.var".
+
+alias id "Ha" = "cic:/CoRN/ftc/Rolle/Rolle/Ha.var".
+
+alias id "Hb" = "cic:/CoRN/ftc/Rolle/Rolle/Hb.var".
+
+(* end hide *)
+
+(* begin show *)
+
+alias id "Fab" = "cic:/CoRN/ftc/Rolle/Rolle/Fab.var".
+
+(* end show *)
+
+(* begin hide *)
+
+alias id "e" = "cic:/CoRN/ftc/Rolle/Rolle/e.var".
+
+alias id "He" = "cic:/CoRN/ftc/Rolle/Rolle/He.var".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/contF'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/derivF.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma2.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/df.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hdf.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hf.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma3.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/df'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hdf'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hf'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/d.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hd.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/incF.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/n.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/fcp.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma1.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/incF'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/fcp'.con" "Rolle__".
+
+(* NOTATION
+Notation cp := (compact_part a b Hab' d Hd).
+*)
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma4.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma5.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma6.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma7.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/j.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hj.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hj'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/k.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hk.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Hk'.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma8.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma9.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma10.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma11.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma12.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma13.con" "Rolle__".
+
+inline "cic:/CoRN/ftc/Rolle/Rolle/Rolle_lemma15.con" "Rolle__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Rolle/Rolle.con".
+
+(* UNEXPORTED
+End Rolle
+*)
+
+(* UNEXPORTED
+Section Law_of_the_Mean
+*)
+
+(*#*
+The following is a simple corollary:
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/b.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/Hab'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/Hab.con" "Law_of_the_Mean__".
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/I.con" "Law_of_the_Mean__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/F.var".
+
+alias id "F'" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/F'.var".
+
+alias id "HF" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HF.var".
+
+(* begin show *)
+
+alias id "HA" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HA.var".
+
+alias id "HB" = "cic:/CoRN/ftc/Rolle/Law_of_the_Mean/HB.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean_I.con".
+
+(* UNEXPORTED
+End Law_of_the_Mean
+*)
+
+(* UNEXPORTED
+Section Corollaries
+*)
+
+(*#*
+We can also state these theorems without expliciting the derivative of [F].
+*)
+
+alias id "a" = "cic:/CoRN/ftc/Rolle/Corollaries/a.var".
+
+alias id "b" = "cic:/CoRN/ftc/Rolle/Corollaries/b.var".
+
+alias id "Hab'" = "cic:/CoRN/ftc/Rolle/Corollaries/Hab'.var".
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Rolle/Corollaries/Hab.con" "Corollaries__".
+
+(* end hide *)
+
+alias id "F" = "cic:/CoRN/ftc/Rolle/Corollaries/F.var".
+
+(* begin show *)
+
+alias id "HF" = "cic:/CoRN/ftc/Rolle/Corollaries/HF.var".
+
+(* end show *)
+
+inline "cic:/CoRN/ftc/Rolle/Rolle'.con".
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean'_I.con".
+
+(* UNEXPORTED
+End Corollaries
+*)
+
+(* UNEXPORTED
+Section Generalizations
+*)
+
+(*#*
+The mean law is more useful if we abstract [a] and [b] from the
+context---allowing them in particular to be equal. In the case where
+[F(a) [=] F(b)] we get Rolle's theorem again, so there is no need to
+state it also in this form.
+
+%\begin{convention}% Assume [I] is a proper interval, [F,F':PartIR].
+%\end{convention}%
+*)
+
+alias id "I" = "cic:/CoRN/ftc/Rolle/Generalizations/I.var".
+
+alias id "pI" = "cic:/CoRN/ftc/Rolle/Generalizations/pI.var".
+
+alias id "F" = "cic:/CoRN/ftc/Rolle/Generalizations/F.var".
+
+alias id "F'" = "cic:/CoRN/ftc/Rolle/Generalizations/F'.var".
+
+(* begin show *)
+
+alias id "derF" = "cic:/CoRN/ftc/Rolle/Generalizations/derF.var".
+
+(* end show *)
+
+(* begin hide *)
+
+inline "cic:/CoRN/ftc/Rolle/Generalizations/incF.con" "Generalizations__".
+
+inline "cic:/CoRN/ftc/Rolle/Generalizations/incF'.con" "Generalizations__".
+
+(* end hide *)
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean.con".
+
+(*#*
+We further generalize the mean law by writing as an explicit bound.
+*)
+
+inline "cic:/CoRN/ftc/Rolle/Law_of_the_Mean_ineq.con".
+
+(* UNEXPORTED
+End Generalizations
+*)
+
projects / helm.git / blobdiff