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[helm.git] / matita / contribs / CoRN-Decl / metrics / ContFunctions.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/metrics/ContFunctions".
+
+include "CoRN.ma".
+
+(* $Id: ContFunctions.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
+
+include "metrics/CPseudoMSpaces.ma".
+
+(* UNEXPORTED
+Section Continuous_functions
+*)
+
+(*#* **Continuous functions, uniformly continuous functions and Lipschitz functions
+%\begin{convention}%
+Let [A] and [B] be pseudo metric spaces.
+%\end{convention}%
+*)
+
+alias id "A" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/A.var".
+
+alias id "B" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/B.var".
+
+(*#*
+We will look at some notions of continuous functions.
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/continuous'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous''.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/lipschitz.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/lipschitz'.con".
+
+(* UNEXPORTED
+End Continuous_functions
+*)
+
+(* UNEXPORTED
+Implicit Arguments continuous [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments uni_continuous [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments lipschitz [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments continuous' [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments uni_continuous' [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments uni_continuous'' [A B].
+*)
+
+(* UNEXPORTED
+Implicit Arguments lipschitz' [A B].
+*)
+
+(* UNEXPORTED
+Section Lemmas
+*)
+
+(* begin hide *)
+
+inline "cic:/CoRN/metrics/ContFunctions/nexp_power.con".
+
+(* end hide *)
+
+inline "cic:/CoRN/metrics/ContFunctions/continuous_imp_continuous'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/continuous'_imp_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_uni_continuous'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous''.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_lipschitz'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/lipschitz'_imp_lipschitz.con".
+
+(*#*
+Every uniformly continuous function is continuous and 
+every Lipschitz function is uniformly continuous.
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_uni_continuous.con".
+
+(* UNEXPORTED
+End Lemmas
+*)
+
+(* UNEXPORTED
+Section Identity
+*)
+
+(*#* **Identity 
+*)
+
+(*#*
+The identity function is Lipschitz. 
+Hence it is uniformly continuous and continuous.
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/id_is_lipschitz.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/id_is_uni_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/id_is_continuous.con".
+
+(* UNEXPORTED
+End Identity
+*)
+
+(* UNEXPORTED
+Section Constant
+*)
+
+(*#* **Constant functions
+%\begin{convention}%
+Let [B] and [X] be pseudo metric spaces.
+%\end{convention}%
+*)
+
+(*#*
+Any constant function is Lipschitz. 
+Hence it is uniformly continuous and continuous.
+*)
+
+alias id "B" = "cic:/CoRN/metrics/ContFunctions/Constant/B.var".
+
+alias id "X" = "cic:/CoRN/metrics/ContFunctions/Constant/X.var".
+
+inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_lipschitz.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_uni_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/const_fun_is_continuous.con".
+
+(* UNEXPORTED
+End Constant
+*)
+
+(* UNEXPORTED
+Section Composition
+*)
+
+(*#* **Composition
+%\begin{convention}%
+Let [B],[C] and [X] be pseudo metric spaces.
+Let [f : (CSetoid_fun X B)] and
+[g : (CSetoid_fun  B C)].
+%\end{convention}%
+*)
+
+(*#*
+The composition of two Lipschitz/uniformly continous/continuous functions is 
+again Lipschitz/uniformly continuous/continuous.
+*)
+
+alias id "X" = "cic:/CoRN/metrics/ContFunctions/Composition/X.var".
+
+alias id "B" = "cic:/CoRN/metrics/ContFunctions/Composition/B.var".
+
+alias id "f" = "cic:/CoRN/metrics/ContFunctions/Composition/f.var".
+
+alias id "C" = "cic:/CoRN/metrics/ContFunctions/Composition/C.var".
+
+alias id "g" = "cic:/CoRN/metrics/ContFunctions/Composition/g.var".
+
+inline "cic:/CoRN/metrics/ContFunctions/comp_resp_lipschitz.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/comp_resp_uni_continuous.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/comp_resp_continuous.con".
+
+(* UNEXPORTED
+End Composition
+*)
+
+(* UNEXPORTED
+Section Limit
+*)
+
+(*#* **Limit
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit.con".
+
+(* UNEXPORTED
+Implicit Arguments MSseqLimit [X].
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit'.con".
+
+(* UNEXPORTED
+Implicit Arguments MSseqLimit' [X].
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit_imp_MSseqLimit'.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/MSseqLimit'_imp_MSseqLimit.con".
+
+inline "cic:/CoRN/metrics/ContFunctions/seqcontinuous'.con".
+
+(* UNEXPORTED
+Implicit Arguments seqcontinuous' [A B].
+*)
+
+inline "cic:/CoRN/metrics/ContFunctions/continuous'_imp_seqcontinuous'.con".
+
+(* UNEXPORTED
+End Limit
+*)
+