1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: IR_CPMSpace.v,v 1.4 2004/04/23 10:01:02 lcf Exp $ *)
21 include "metrics/ContFunctions.ma".
32 Let [X] be a pseudo metric space.
37 The real numbers with the usual distance form a pseudo metric space.
40 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR.con" as definition.
42 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/bin_fun_strext_dIR.con" as lemma.
44 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR_as_CSetoid_fun.con" as definition.
46 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR_nneg.con" as lemma.
48 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR_com.con" as lemma.
50 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR_pos_imp_ap.con" as lemma.
54 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/IR_tri_ineq.con" as lemma.
58 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/dIR_tri_ineq.con" as lemma.
60 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/IR_dIR_is_CPsMetricSpace.con" as definition.
62 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/IR_as_CPsMetricSpace.con" as definition.
65 cic:/CoRN/metrics/IR_CPMSpace/Reals/X.var
68 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/rev_tri_ineq'.con" as lemma.
71 A pseudo metric is Lipschitz. Hence it is uniformly continuous and continuous.
74 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/d_is_lipschitz.con" as lemma.
76 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/d_is_uni_continuous.con" as lemma.
78 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/d_is_continuous.con" as lemma.
88 (*#* **Addition of continuous functions
92 The sum of two Lipschitz/uniformly continous/continuous functions is again
93 Lipschitz/uniformly continuous/continuous.
96 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/plus_resp_lipschitz.con" as lemma.
98 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/plus_resp_uni_continuous.con" as lemma.
100 inline procedural "cic:/CoRN/metrics/IR_CPMSpace/plus_resp_continuous.con" as lemma.