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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: Prod_Sub.v,v 1.4 2004/04/23 10:01:02 lcf Exp $ *)
21 include "metrics/IR_CPMSpace.ma".
27 (*#* **Product-Pseudo-Metric-Spaces
31 The product metric here defined is:
32 $ d_{prod}((a_1,b_1),(a_2,b_2)):= d_A(a_1,a_2)+d_B(b_1,b_2)$
33 # d<SUB>prod</SUB>((a<SUB>1</SUB>,b<SUB>1</SUB>),(a<SUB>2</SUB>,b<SUB>2</SUB>)):= d<SUB>A</SUB>(a<SUB>1</SUB>,b<SUB>1</SUB>)+d<SUB>B</SUB>(b<SUB>1</SUB>,b<SUB>2</SUB>)#.
34 This is %\emph{not}% #<I>not</I># the one used to make the metric of
35 $\RR^{2}$ #IR<SUP>2</SUP># out of the metric of $\RR$ #IR#.
38 inline procedural "cic:/CoRN/metrics/Prod_Sub/dprod0.con" as definition.
40 inline procedural "cic:/CoRN/metrics/Prod_Sub/dprod0_strext.con" as lemma.
42 inline procedural "cic:/CoRN/metrics/Prod_Sub/d_prod0.con" as definition.
44 inline procedural "cic:/CoRN/metrics/Prod_Sub/prod0cpsmetricspace_is_CPsMetricSpace.con" as lemma.
46 inline procedural "cic:/CoRN/metrics/Prod_Sub/Prod0CPsMetricSpace.con" as definition.
56 (*#* **Sub-Pseudo-Metric-Spaces
60 The pseudo metric on a subspace $Y$ #Y# of a pseudo metric space $X$ #X# is
61 the pseudo metric on $X$ #X# restricted to $Y$ #Y#.
64 inline procedural "cic:/CoRN/metrics/Prod_Sub/restr_bin_fun.con" as definition.
67 Implicit Arguments restr_bin_fun [X].
70 inline procedural "cic:/CoRN/metrics/Prod_Sub/restr_bin_fun'.con" as definition.
73 Implicit Arguments restr_bin_fun' [X].
76 inline procedural "cic:/CoRN/metrics/Prod_Sub/restr_bin_fun_strext.con" as lemma.
78 inline procedural "cic:/CoRN/metrics/Prod_Sub/Build_SubCSetoid_bin_fun.con" as definition.
80 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub.con" as definition.
83 Implicit Arguments dsub [X].
86 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub_com.con" as lemma.
88 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub_nneg.con" as lemma.
90 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub_pos_imp_ap.con" as lemma.
92 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub_tri_ineq.con" as lemma.
94 inline procedural "cic:/CoRN/metrics/Prod_Sub/is_SubPsMetricSpace.con" as definition.
96 inline procedural "cic:/CoRN/metrics/Prod_Sub/SubPsMetricSpace.con" as definition.
99 Implicit Arguments SubPsMetricSpace [X].
102 inline procedural "cic:/CoRN/metrics/Prod_Sub/from_SubPsMetricSpace.con" as definition.
105 The function [dsub'] is used in the definition of %''located''% #"located"#.
106 It enables one to speak about a distance between an element of a
107 pseudo metric space and a certain subspace.
110 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub'.con" as definition.
113 Implicit Arguments dsub' [X].
116 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub'_strext.con" as lemma.
118 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub'_as_cs_fun.con" as definition.
121 Implicit Arguments dsub'_as_cs_fun [X].
124 inline procedural "cic:/CoRN/metrics/Prod_Sub/dsub'_uni_continuous''.con" as lemma.