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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 *)
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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: ContFunctions.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
21 include "metrics/CPseudoMSpaces.ma".
24 Section Continuous_functions
27 (*#* **Continuous functions, uniformly continuous functions and Lipschitz functions
29 Let [A] and [B] be pseudo metric spaces.
33 alias id "A" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/A.var".
35 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Continuous_functions/B.var".
38 We will look at some notions of continuous functions.
41 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous.con" as definition.
43 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'.con" as definition.
45 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous.con" as definition.
47 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'.con" as definition.
49 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous''.con" as definition.
51 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz.con" as definition.
53 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz'.con" as definition.
56 End Continuous_functions
60 Implicit Arguments continuous [A B].
64 Implicit Arguments uni_continuous [A B].
68 Implicit Arguments lipschitz [A B].
72 Implicit Arguments continuous' [A B].
76 Implicit Arguments uni_continuous' [A B].
80 Implicit Arguments uni_continuous'' [A B].
84 Implicit Arguments lipschitz' [A B].
93 inline procedural "cic:/CoRN/metrics/ContFunctions/nexp_power.con" as lemma.
97 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous_imp_continuous'.con" as lemma.
99 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'_imp_continuous.con" as lemma.
101 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_uni_continuous'.con" as lemma.
103 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous.con" as lemma.
105 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous''.con" as lemma.
107 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_lipschitz'.con" as lemma.
109 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz'_imp_lipschitz.con" as lemma.
112 Every uniformly continuous function is continuous and
113 every Lipschitz function is uniformly continuous.
116 inline procedural "cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_continuous.con" as lemma.
118 inline procedural "cic:/CoRN/metrics/ContFunctions/lipschitz_imp_uni_continuous.con" as lemma.
132 The identity function is Lipschitz.
133 Hence it is uniformly continuous and continuous.
136 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_lipschitz.con" as lemma.
138 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_uni_continuous.con" as lemma.
140 inline procedural "cic:/CoRN/metrics/ContFunctions/id_is_continuous.con" as lemma.
150 (*#* **Constant functions
152 Let [B] and [X] be pseudo metric spaces.
157 Any constant function is Lipschitz.
158 Hence it is uniformly continuous and continuous.
161 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Constant/B.var".
163 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Constant/X.var".
165 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_lipschitz.con" as lemma.
167 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_uni_continuous.con" as lemma.
169 inline procedural "cic:/CoRN/metrics/ContFunctions/const_fun_is_continuous.con" as lemma.
181 Let [B],[C] and [X] be pseudo metric spaces.
182 Let [f : (CSetoid_fun X B)] and
183 [g : (CSetoid_fun B C)].
188 The composition of two Lipschitz/uniformly continous/continuous functions is
189 again Lipschitz/uniformly continuous/continuous.
192 alias id "X" = "cic:/CoRN/metrics/ContFunctions/Composition/X.var".
194 alias id "B" = "cic:/CoRN/metrics/ContFunctions/Composition/B.var".
196 alias id "f" = "cic:/CoRN/metrics/ContFunctions/Composition/f.var".
198 alias id "C" = "cic:/CoRN/metrics/ContFunctions/Composition/C.var".
200 alias id "g" = "cic:/CoRN/metrics/ContFunctions/Composition/g.var".
202 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_lipschitz.con" as lemma.
204 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_uni_continuous.con" as lemma.
206 inline procedural "cic:/CoRN/metrics/ContFunctions/comp_resp_continuous.con" as lemma.
219 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit.con" as definition.
222 Implicit Arguments MSseqLimit [X].
225 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit'.con" as definition.
228 Implicit Arguments MSseqLimit' [X].
231 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit_imp_MSseqLimit'.con" as lemma.
233 inline procedural "cic:/CoRN/metrics/ContFunctions/MSseqLimit'_imp_MSseqLimit.con" as lemma.
235 inline procedural "cic:/CoRN/metrics/ContFunctions/seqcontinuous'.con" as definition.
238 Implicit Arguments seqcontinuous' [A B].
241 inline procedural "cic:/CoRN/metrics/ContFunctions/continuous'_imp_seqcontinuous'.con" as lemma.