Fiddling to get clean rendered version
authorjcheney@inf.ed.ac.uk
Mon, 25 Feb 2013 15:15:58 +0000
changeset 5735 8c51d3082048
parent 5734 8df8ca08c88d
child 5736 0ff2dc0bbac3
Fiddling to get clean rendered version
semantics/prov-sem-review-20130226.html
semantics/prov-sem.html
--- a/semantics/prov-sem-review-20130226.html	Mon Feb 25 13:49:34 2013 +0000
+++ b/semantics/prov-sem-review-20130226.html	Mon Feb 25 15:15:58 2013 +0000
@@ -584,10 +584,7 @@
     
 
   
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+    <style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
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+#MathJax_Message {position: fixed; left: 1px; bottom: 2px; background-color: #E6E6E6; border: 1px solid #959595; margin: 0px; padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto; white-space: nowrap}
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  * ReSpec 3 CSS
  * Robin Berjon - http://berjon.com/
@@ -763,29 +781,11 @@
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-  <body><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div class="head">
+</style><link href="https://www.w3.org/StyleSheets/TR/W3C-ED" rel="stylesheet"><!--[if lt IE 9]><script src='https://www.w3.org/2008/site/js/html5shiv.js'></script><![endif]--><!--[if lt IE 9]><script src='https://www.w3.org/2008/site/js/html5shiv.js'></script><![endif]--></head> 
+  <body><div class="head">
   <p>
     
-      <a href="http://www.w3.org/"><img width="72" height="48" src="http://www.w3.org/Icons/w3c_home" alt="W3C"></a>
+      <a href="http://www.w3.org/"><img src="https://www.w3.org/Icons/w3c_home" alt="W3C" height="48" width="72"></a>
     
   </p>
   <h1 class="title" id="title">Semantics of the PROV Data Model</h1>
@@ -834,9 +834,9 @@
     
   
   <hr>
-</div><div id="MathJax_Message" style="display: none;"></div> 
-
-    <section id="abstract" class="introductory"><h2>Abstract</h2>
+</div><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px none; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div style="display: none;" id="MathJax_Message"></div> 
+
+    <section class="introductory" id="abstract"><h2>Abstract</h2>
 <p>
 Provenance is information about entities, activities, and people
 involved in producing a piece of data or thing, which can be used to
@@ -875,20 +875,20 @@
         
 <h4 id="prov-family-of-documents">PROV Family of Documents</h4>
 This document is part of the PROV family of documents, a set of documents defining various aspects that are necessary to achieve the vision of inter-operable
-interchange of provenance information in heterogeneous environments such as the Web.  These documents are listed below. Please consult the [<cite><a class="bibref" href="#bib-PROV-OVERVIEW">PROV-OVERVIEW</a></cite>] for a guide to reading these documents. 
+interchange of provenance information in heterogeneous environments such as the Web.  These documents are listed below. Please consult the [<cite><a href="#bib-PROV-OVERVIEW" class="bibref">PROV-OVERVIEW</a></cite>] for a guide to reading these documents. 
 <ul>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-overview-20130312/">PROV-OVERVIEW</a> (To be published as Note), an overview of the PROV family of documents [<cite><a class="bibref" href="#bib-PROV-OVERVIEW">PROV-OVERVIEW</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-primer-20130312/">PROV-PRIMER</a> (To be published as Note), a primer for the PROV data model [<cite><a class="bibref" href="#bib-PROV-PRIMER">PROV-PRIMER</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-o-20130312/">PROV-O</a> (Proposed Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [<cite><a class="bibref" href="#bib-PROV-O">PROV-O</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-dm-20130312/">PROV-DM</a> (Proposed Recommendation), the PROV data model for provenance [<cite><a class="bibref" href="#bib-PROV-DM">PROV-DM</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/PR-prov-n-20130312/">PROV-N</a> (Proposed Recommendation), a notation for provenance aimed at human consumption [<cite><a class="bibref" href="#bib-PROV-N">PROV-N</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-overview-20130312/">PROV-OVERVIEW</a> (To be published as Note), an overview of the PROV family of documents [<cite><a href="#bib-PROV-OVERVIEW" class="bibref">PROV-OVERVIEW</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-primer-20130312/">PROV-PRIMER</a> (To be published as Note), a primer for the PROV data model [<cite><a href="#bib-PROV-PRIMER" class="bibref">PROV-PRIMER</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/PR-prov-o-20130312/">PROV-O</a> (Proposed Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [<cite><a href="#bib-PROV-O" class="bibref">PROV-O</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/PR-prov-dm-20130312/">PROV-DM</a> (Proposed Recommendation), the PROV data model for provenance [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/PR-prov-n-20130312/">PROV-N</a> (Proposed Recommendation), a notation for provenance aimed at human consumption [<cite><a href="#bib-PROV-N" class="bibref">PROV-N</a></cite>];</li>
 <li> <a href="http://www.w3.org/TR/2013/PR-prov-constraints-20130312/">PROV-CONSTRAINTS</a> (Proposed Recommendation), a set of constraints applying to the PROV data model  (this document);</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-xml-20130312/">PROV-XML</a> (To be published as Note),  an XML schema for the PROV data model [<cite><a class="bibref" href="#bib-PROV-XML">PROV-XML</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-aq-20130312/">PROV-AQ</a> (To be published as Note), the mechanisms for accessing and querying provenance [<cite><a class="bibref" href="#bib-PROV-AQ">PROV-AQ</a></cite>]; </li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-dictionary-20130312/">PROV-DICTIONARY</a> (To be published as Note) introduces a specific type of collection, consisting of key-entity pairs [<cite><a class="bibref" href="#bib-PROV-DICTIONARY">PROV-DICTIONARY</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-dc-20130312/">PROV-DC</a> (To be published as Note) provides a mapping between PROV and Dublic Core Terms [<cite><a class="bibref" href="#bib-PROV-DC">PROV-DC</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-sem-20130312/">PROV-SEM</a> (To be published as Note), a declarative specification in terms of first-order logic of the PROV data model [<cite><a class="bibref" href="#bib-PROV-SEM">PROV-SEM</a></cite>];</li>
-<li> <a href="http://www.w3.org/TR/2013/WD-prov-links-20130312/">PROV-LINKS</a> (To be published as Note) introduces a mechanism to link across bundles [<cite><a class="bibref" href="#bib-PROV-LINKS">PROV-LINKS</a></cite>].</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-xml-20130312/">PROV-XML</a> (To be published as Note),  an XML schema for the PROV data model [<cite><a href="#bib-PROV-XML" class="bibref">PROV-XML</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-aq-20130312/">PROV-AQ</a> (To be published as Note), the mechanisms for accessing and querying provenance [<cite><a href="#bib-PROV-AQ" class="bibref">PROV-AQ</a></cite>]; </li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-dictionary-20130312/">PROV-DICTIONARY</a> (To be published as Note) introduces a specific type of collection, consisting of key-entity pairs [<cite><a href="#bib-PROV-DICTIONARY" class="bibref">PROV-DICTIONARY</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-dc-20130312/">PROV-DC</a> (To be published as Note) provides a mapping between PROV and Dublic Core Terms [<cite><a href="#bib-PROV-DC" class="bibref">PROV-DC</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-sem-20130312/">PROV-SEM</a> (To be published as Note), a declarative specification in terms of first-order logic of the PROV data model [<cite><a href="#bib-PROV-SEM" class="bibref">PROV-SEM</a></cite>];</li>
+<li> <a href="http://www.w3.org/TR/2013/WD-prov-links-20130312/">PROV-LINKS</a> (To be published as Note) introduces a mechanism to link across bundles [<cite><a href="#bib-PROV-LINKS" class="bibref">PROV-LINKS</a></cite>].</li>
 </ul>
 
 
@@ -936,7 +936,7 @@
       
     
   
-</section><section id="toc"><h2 class="introductory">Table of Contents</h2><ul class="toc"><li class="tocline"><a href="#introduction" class="tocxref"><span class="secno">1. </span>Introduction</a><ul class="toc"><li class="tocline"><a href="#purpose" class="tocxref"><span class="secno">1.1 </span>Purpose of this document</a></li><li class="tocline"><a href="#structure-of-this-document" class="tocxref"><span class="secno">1.2 </span>Structure of this document</a></li><li class="tocline"><a href="#audience" class="tocxref"><span class="secno">1.3 </span> Audience </a></li></ul></li><li class="tocline"><a href="#basics" class="tocxref"><span class="secno">2. </span> Basics </a><ul class="toc"><li class="tocline"><a href="#identifiers" class="tocxref"><span class="secno">2.1 </span> Identifiers </a></li><li class="tocline"><a href="#times-and-intervals" class="tocxref"><span class="secno">2.2 </span> Times and Intervals </a></li><li class="tocline"><a href="#attributes-and-values" class="tocxref"><span class="secno">2.3 </span> Attributes and Values </a></li><li class="tocline"><a href="#formulas" class="tocxref"><span class="secno">2.4 </span>Atomic Formulas</a></li><li class="tocline"><a href="#first-order-formulas" class="tocxref"><span class="secno">2.5 </span>First-Order Formulas</a></li></ul></li><li class="tocline"><a href="#structures" class="tocxref"><span class="secno">3. </span> Structures and Interpretations </a><ul class="toc"><li class="tocline"><a href="#things-1" class="tocxref"><span class="secno">3.1 </span> Things </a></li><li class="tocline"><a href="#objects-1" class="tocxref"><span class="secno">3.2 </span> Objects </a><ul class="toc"><li class="tocline"><a href="#entities-1" class="tocxref"><span class="secno">3.2.1 </span> Entities </a><ul class="toc"><li class="tocline"><a href="#plans-1" class="tocxref"><span class="secno">3.2.1.1 </span> Plans </a></li></ul></li><li class="tocline"><a href="#actvities" class="tocxref"><span class="secno">3.2.2 </span> Actvities </a></li><li class="tocline"><a href="#agents-1" class="tocxref"><span class="secno">3.2.3 </span> Agents </a></li><li class="tocline"><a href="#interactions-1" class="tocxref"><span class="secno">3.2.4 </span> Interactions </a><ul class="toc"><li class="tocline"><a href="#events-1" class="tocxref"><span class="secno">3.2.4.1 </span> Events </a></li><li class="tocline"><a href="#associations-1" class="tocxref"><span class="secno">3.2.4.2 </span> Associations </a></li><li class="tocline"><a href="#communications" class="tocxref"><span class="secno">3.2.4.3 </span>Communications</a></li><li class="tocline"><a href="#delegations" class="tocxref"><span class="secno">3.2.4.4 </span>Delegations</a></li><li class="tocline"><a href="#derivations-1" class="tocxref"><span class="secno">3.2.4.5 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a href="#relations" class="tocxref"><span class="secno">3.3 </span> Relations </a><ul class="toc"><li class="tocline"><a href="#derivation-paths-and-derivedfrom" class="tocxref"><span class="secno">3.3.1 </span> Derivation paths and DerivedFrom </a></li></ul></li><li class="tocline"><a href="#putting-it-all-together" class="tocxref"><span class="secno">3.4 </span> Putting it all together </a></li><li class="tocline"><a href="#interpretations" class="tocxref"><span class="secno">3.5 </span> Interpretations </a></li></ul></li><li class="tocline"><a href="#semantics" class="tocxref"><span class="secno">4. </span> Semantics </a><ul class="toc"><li class="tocline"><a href="#satisfaction" class="tocxref"><span class="secno">4.1 </span> Satisfaction </a></li><li class="tocline"><a href="#attribute-matching" class="tocxref"><span class="secno">4.2 </span> Attribute matching </a></li><li class="tocline"><a href="#semantics-of-element-formulas" class="tocxref"><span class="secno">4.3 </span> Semantics of Element Formulas </a><ul class="toc"><li class="tocline"><a href="#entity" class="tocxref"><span class="secno">4.3.1 </span> Entity </a></li><li class="tocline"><a href="#activity" class="tocxref"><span class="secno">4.3.2 </span> Activity </a></li><li class="tocline"><a href="#agent" class="tocxref"><span class="secno">4.3.3 </span> Agent </a></li></ul></li><li class="tocline"><a href="#semantics-of-relations" class="tocxref"><span class="secno">4.4 </span> Semantics of Relations </a><ul class="toc"><li class="tocline"><a href="#generation" class="tocxref"><span class="secno">4.4.1 </span> Generation </a></li><li class="tocline"><a href="#use" class="tocxref"><span class="secno">4.4.2 </span> Use </a></li><li class="tocline"><a href="#invalidation" class="tocxref"><span class="secno">4.4.3 </span> Invalidation </a></li><li class="tocline"><a href="#association" class="tocxref"><span class="secno">4.4.4 </span> Association </a></li><li class="tocline"><a href="#start-formulas" class="tocxref"><span class="secno">4.4.5 </span> Start Formulas </a></li><li class="tocline"><a href="#end" class="tocxref"><span class="secno">4.4.6 </span> End </a></li><li class="tocline"><a href="#attribution" class="tocxref"><span class="secno">4.4.7 </span> Attribution </a></li><li class="tocline"><a href="#communication" class="tocxref"><span class="secno">4.4.8 </span>Communication</a></li><li class="tocline"><a href="#responsibility" class="tocxref"><span class="secno">4.4.9 </span> Responsibility </a></li><li class="tocline"><a href="#derivation" class="tocxref"><span class="secno">4.4.10 </span> Derivation </a><ul class="toc"><li class="tocline"><a href="#precise" class="tocxref"><span class="secno">4.4.10.1 </span> Precise </a></li><li class="tocline"><a href="#imprecise" class="tocxref"><span class="secno">4.4.10.2 </span> Imprecise </a></li></ul></li><li class="tocline"><a href="#influence" class="tocxref"><span class="secno">4.4.11 </span>Influence</a></li><li class="tocline"><a href="#specialization" class="tocxref"><span class="secno">4.4.12 </span> Specialization </a></li><li class="tocline"><a href="#alternate" class="tocxref"><span class="secno">4.4.13 </span> Alternate </a></li><li class="tocline"><a href="#membership" class="tocxref"><span class="secno">4.4.14 </span> Membership </a></li></ul></li><li class="tocline"><a href="#auxiliary-formulas" class="tocxref"><span class="secno">4.5 </span>Auxiliary formulas</a><ul class="toc"><li class="tocline"><a href="#equals" class="tocxref"><span class="secno">4.5.1 </span>Equals</a></li><li class="tocline"><a href="#precedes-and-strictly-precedes" class="tocxref"><span class="secno">4.5.2 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a href="#notnull" class="tocxref"><span class="secno">4.5.3 </span>notNull</a></li><li class="tocline"><a href="#typeof" class="tocxref"><span class="secno">4.5.4 </span>typeOf</a></li></ul></li></ul></li><li class="tocline"><a href="#theory" class="tocxref"><span class="secno">5. </span> Inferences and Constraints </a><ul class="toc"><li class="tocline"><a href="#inferences" class="tocxref"><span class="secno">5.1 </span>Inferences</a></li><li class="tocline"><a href="#constraints" class="tocxref"><span class="secno">5.2 </span>Constraints</a><ul class="toc"><li class="tocline"><a href="#uniqueness-constraints" class="tocxref"><span class="secno">5.2.1 </span>Uniqueness constraints</a></li><li class="tocline"><a href="#ordering-constraints" class="tocxref"><span class="secno">5.2.2 </span>Ordering constraints</a></li><li class="tocline"><a href="#typing-constraints" class="tocxref"><span class="secno">5.2.3 </span>Typing constraints</a></li><li class="tocline"><a href="#impossibility-constraints" class="tocxref"><span class="secno">5.2.4 </span>Impossibility constraints</a></li></ul></li></ul></li><li class="tocline"><a href="#soundness" class="tocxref"><span class="secno">6. </span>Soundness</a></li><li class="tocline"><a href="#acknowledgements" class="tocxref"><span class="secno">A. </span>Acknowledgements</a></li><li class="tocline"><a href="#references" class="tocxref"><span class="secno">B. </span>References</a><ul class="toc"><li class="tocline"><a href="#informative-references" class="tocxref"><span class="secno">B.1 </span>Informative references</a></li></ul></li></ul></section>
+</section><section id="toc"><h2 class="introductory">Table of Contents</h2><ul class="toc"><li class="tocline"><a class="tocxref" href="#introduction"><span class="secno">1. </span>Introduction</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#purpose"><span class="secno">1.1 </span>Purpose of this document</a></li><li class="tocline"><a class="tocxref" href="#structure-of-this-document"><span class="secno">1.2 </span>Structure of this document</a></li><li class="tocline"><a class="tocxref" href="#audience"><span class="secno">1.3 </span> Audience </a></li></ul></li><li class="tocline"><a class="tocxref" href="#basics"><span class="secno">2. </span> Basics </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#identifiers"><span class="secno">2.1 </span> Identifiers </a></li><li class="tocline"><a class="tocxref" href="#times-and-intervals"><span class="secno">2.2 </span> Times and Intervals </a></li><li class="tocline"><a class="tocxref" href="#attributes-and-values"><span class="secno">2.3 </span> Attributes and Values </a></li><li class="tocline"><a class="tocxref" href="#formulas"><span class="secno">2.4 </span>Atomic Formulas</a></li><li class="tocline"><a class="tocxref" href="#first-order-formulas"><span class="secno">2.5 </span>First-Order Formulas</a></li></ul></li><li class="tocline"><a class="tocxref" href="#structures"><span class="secno">3. </span> Structures and Interpretations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#things-1"><span class="secno">3.1 </span> Things </a></li><li class="tocline"><a class="tocxref" href="#objects-1"><span class="secno">3.2 </span> Objects </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#entities-1"><span class="secno">3.2.1 </span> Entities </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#plans-1"><span class="secno">3.2.1.1 </span> Plans </a></li></ul></li><li class="tocline"><a class="tocxref" href="#actvities"><span class="secno">3.2.2 </span> Actvities </a></li><li class="tocline"><a class="tocxref" href="#agents-1"><span class="secno">3.2.3 </span> Agents </a></li><li class="tocline"><a class="tocxref" href="#interactions-1"><span class="secno">3.2.4 </span> Interactions </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#events-1"><span class="secno">3.2.4.1 </span> Events </a></li><li class="tocline"><a class="tocxref" href="#associations-1"><span class="secno">3.2.4.2 </span> Associations </a></li><li class="tocline"><a class="tocxref" href="#communications"><span class="secno">3.2.4.3 </span>Communications</a></li><li class="tocline"><a class="tocxref" href="#delegations"><span class="secno">3.2.4.4 </span>Delegations</a></li><li class="tocline"><a class="tocxref" href="#derivations-1"><span class="secno">3.2.4.5 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#relations"><span class="secno">3.3 </span> Relations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#derivation-paths-and-derivedfrom"><span class="secno">3.3.1 </span> Derivation paths and DerivedFrom </a></li></ul></li><li class="tocline"><a class="tocxref" href="#putting-it-all-together"><span class="secno">3.4 </span> Putting it all together </a></li><li class="tocline"><a class="tocxref" href="#interpretations"><span class="secno">3.5 </span> Interpretations </a></li></ul></li><li class="tocline"><a class="tocxref" href="#semantics"><span class="secno">4. </span> Semantics </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#satisfaction"><span class="secno">4.1 </span> Satisfaction </a></li><li class="tocline"><a class="tocxref" href="#attribute-matching"><span class="secno">4.2 </span> Attribute matching </a></li><li class="tocline"><a class="tocxref" href="#semantics-of-element-formulas"><span class="secno">4.3 </span> Semantics of Element Formulas </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#entity"><span class="secno">4.3.1 </span> Entity </a></li><li class="tocline"><a class="tocxref" href="#activity"><span class="secno">4.3.2 </span> Activity </a></li><li class="tocline"><a class="tocxref" href="#agent"><span class="secno">4.3.3 </span> Agent </a></li></ul></li><li class="tocline"><a class="tocxref" href="#semantics-of-relations"><span class="secno">4.4 </span> Semantics of Relations </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#generation"><span class="secno">4.4.1 </span> Generation </a></li><li class="tocline"><a class="tocxref" href="#use"><span class="secno">4.4.2 </span> Use </a></li><li class="tocline"><a class="tocxref" href="#invalidation"><span class="secno">4.4.3 </span> Invalidation </a></li><li class="tocline"><a class="tocxref" href="#association"><span class="secno">4.4.4 </span> Association </a></li><li class="tocline"><a class="tocxref" href="#start-formulas"><span class="secno">4.4.5 </span> Start Formulas </a></li><li class="tocline"><a class="tocxref" href="#end"><span class="secno">4.4.6 </span> End </a></li><li class="tocline"><a class="tocxref" href="#attribution"><span class="secno">4.4.7 </span> Attribution </a></li><li class="tocline"><a class="tocxref" href="#communication"><span class="secno">4.4.8 </span>Communication</a></li><li class="tocline"><a class="tocxref" href="#responsibility"><span class="secno">4.4.9 </span> Responsibility </a></li><li class="tocline"><a class="tocxref" href="#derivation"><span class="secno">4.4.10 </span> Derivation </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#precise"><span class="secno">4.4.10.1 </span> Precise </a></li><li class="tocline"><a class="tocxref" href="#imprecise"><span class="secno">4.4.10.2 </span> Imprecise </a></li></ul></li><li class="tocline"><a class="tocxref" href="#influence"><span class="secno">4.4.11 </span>Influence</a></li><li class="tocline"><a class="tocxref" href="#specialization"><span class="secno">4.4.12 </span> Specialization </a></li><li class="tocline"><a class="tocxref" href="#alternate"><span class="secno">4.4.13 </span> Alternate </a></li><li class="tocline"><a class="tocxref" href="#membership"><span class="secno">4.4.14 </span> Membership </a></li></ul></li><li class="tocline"><a class="tocxref" href="#auxiliary-formulas"><span class="secno">4.5 </span>Auxiliary formulas</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#equals"><span class="secno">4.5.1 </span>Equals</a></li><li class="tocline"><a class="tocxref" href="#precedes-and-strictly-precedes"><span class="secno">4.5.2 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a class="tocxref" href="#notnull"><span class="secno">4.5.3 </span>notNull</a></li><li class="tocline"><a class="tocxref" href="#typeof"><span class="secno">4.5.4 </span>typeOf</a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#theory"><span class="secno">5. </span> Inferences and Constraints </a><ul class="toc"><li class="tocline"><a class="tocxref" href="#inferences"><span class="secno">5.1 </span>Inferences</a></li><li class="tocline"><a class="tocxref" href="#constraints"><span class="secno">5.2 </span>Constraints</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#uniqueness-constraints"><span class="secno">5.2.1 </span>Uniqueness constraints</a></li><li class="tocline"><a class="tocxref" href="#ordering-constraints"><span class="secno">5.2.2 </span>Ordering constraints</a></li><li class="tocline"><a class="tocxref" href="#typing-constraints"><span class="secno">5.2.3 </span>Typing constraints</a></li><li class="tocline"><a class="tocxref" href="#impossibility-constraints"><span class="secno">5.2.4 </span>Impossibility constraints</a></li></ul></li></ul></li><li class="tocline"><a class="tocxref" href="#soundness"><span class="secno">6. </span>Soundness</a></li><li class="tocline"><a class="tocxref" href="#acknowledgements"><span class="secno">A. </span>Acknowledgements</a></li><li class="tocline"><a class="tocxref" href="#references"><span class="secno">B. </span>References</a><ul class="toc"><li class="tocline"><a class="tocxref" href="#informative-references"><span class="secno">B.1 </span>Informative references</a></li></ul></li></ul></section>
 
 
 
@@ -949,9 +949,9 @@
 <p>
 Provenance is a record that describes the people, institutions, entities, and activities involved in producing, influencing, or delivering a piece of data or a thing.
 This document complements
-  the PROV-DM specification [<cite><a class="bibref" href="#bib-PROV-DM">PROV-DM</a></cite>] that defines a data model for
+  the PROV-DM specification [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>] that defines a data model for
   provenance on the Web, and the PROV-CONSTRAINTS specification
-[<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</a></cite>] that
+[<cite><a href="#bib-PROV-CONSTRAINTS" class="bibref">PROV-CONSTRAINTS</a></cite>] that
 specifies definitions, inferences, and constraints that can be used to
 reason about PROV documents, or determine their validity.  This document
 provides a reference formal semantics of PROV, providing a formal
@@ -986,8 +986,8 @@
 derivation, the meaning is either obvious or situation-dependent.
 However, during the development of PROV, the importance of additional
 concepts became evident, but the intuitive meaning or correct use of
-these concepts were not clear.  For example, the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-1-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1"><span style="display: inline-block; position: relative; width: 91px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-4" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-5" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-6" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-7" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-8" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-9" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-10" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-11" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-12" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-13" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-1">alternateOf</script> and
-<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-2-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-14"><span style="display: inline-block; position: relative; width: 128px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-15"><span class="mi" id="MathJax-Span-16" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-17" style="font-family: STIXGeneral-Italic;">p</span><span class="mi" id="MathJax-Span-18" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-19" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-20" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-21" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-22" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-23" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-24" style="font-family: STIXGeneral-Italic;">z</span><span class="mi" id="MathJax-Span-25" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-26" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-27" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-28" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-29" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-30" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-31" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-2">specializationOf</script> relations are used in PROV to relate different
+these concepts were not clear.  For example, the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-1-Frame" class="MathJax"><nobr><span id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 83px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-3" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-5" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-6" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-7" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-8" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-9" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-10" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-11" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-12" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-13" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-1" type="math/tex">alternateOf</script> and
+<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-2-Frame" class="MathJax"><nobr><span id="MathJax-Span-14" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.51em); top: -2.57em; left: 0em;"><span id="MathJax-Span-15" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-16" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-17" class="mi">p</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-18" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-19" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-20" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-21" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-22" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-23" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-24" class="mi">z</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-25" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-26" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-27" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-28" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-29" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-30" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-31" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-2" type="math/tex">specializationOf</script> relations are used in PROV to relate different
 entities that present aspects of "the same thing".  
 Over time the working group came to a
 consensus about these concepts and how they are to be used, but this
@@ -1044,7 +1044,7 @@
 <div class="note"><div class="note-title"><span>Note</span></div><div class="">
   Completeness direction above doesn't hold for current semantics,
   because attributes assumed to be single-valued; for
-  example, <span class="MathJax_Preview">entity(id,[a=1,a=2])</span><script type="math/tex" id="MathJax-Element-3">entity(id,[a=1,a=2])</script> is valid but doesn't have a
+  example, <span class="MathJax_Preview">entity(id,[a=1,a=2])</span><script id="MathJax-Element-3" type="math/tex">entity(id,[a=1,a=2])</script> is valid but doesn't have a
   reference model.
   </div></div>
 
@@ -1053,10 +1053,10 @@
 <h3><span class="secno">1.3 </span> Audience </h3>
 
 
-<p>This document assumes familiarity with [<cite><a class="bibref" href="#bib-PROV-DM">PROV-DM</a></cite>] and
-  [<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</a></cite>] and employs (a
+<p>This document assumes familiarity with [<cite><a href="#bib-PROV-DM" class="bibref">PROV-DM</a></cite>] and
+  [<cite><a href="#bib-PROV-CONSTRAINTS" class="bibref">PROV-CONSTRAINTS</a></cite>] and employs (a
   simplified form of) 
-[<cite><a class="bibref" href="#bib-PROV-N">PROV-N</a></cite>] notation.  In particular it assumes familiarity with the concepts
+[<cite><a href="#bib-PROV-N" class="bibref">PROV-N</a></cite>] notation.  In particular it assumes familiarity with the concepts
   from logic, and the relationship between PROV statements and
   instances and first-order formulas and theories, respectively,
   presented in <a href="http://www.w3.org/TR/2013/PR-prov-constraints-20130312/#overview">Section 2.5</a> of PROV-CONSTRAINTS.
@@ -1124,37 +1124,37 @@
 <section id="identifiers">
 <h3><span class="secno">2.1 </span> Identifiers </h3>
 
-<p>A lowercase symbol <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-4-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-54"><span style="display: inline-block; position: relative; width: 54px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 55px -9.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-55"><span class="mi" id="MathJax-Span-56" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-57" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-58" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">y</span><span class="mo" id="MathJax-Span-59" style="font-family: STIXGeneral-Regular;">,</span><span class="mo" id="MathJax-Span-60" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">.</span><span class="mo" id="MathJax-Span-61" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">.</span><span class="mo" id="MathJax-Span-62" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">.</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 14.5px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-4">x,y,...</script> on its own denotes an identifier.
-Identifiers may or may not be URIs.  Identifiers are viewed as variables in logic (or blank nodes in RDF): just because we have two different identifiers <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-5-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-63"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -9.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-64"><span class="mi" id="MathJax-Span-65" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-5">x</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-6-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-66"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 55px -9.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-67"><span class="mi" id="MathJax-Span-68" style="font-family: STIXGeneral-Italic;">y</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 14.5px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-6">y</script> doesn't tell us that they denote different things, since we could discover that they are actually the same later.  We write <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-7-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-69"><span style="display: inline-block; position: relative; width: 81px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-70"><span class="mi" id="MathJax-Span-71" style="font-family: STIXGeneral-Italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-72" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mi" id="MathJax-Span-73" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-74" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-75" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-76" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-77" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-78" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-79" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-80" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-81" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-7">Identifiers</script> for the set of identifiers of interest in a given situation (typically, the set of identifiers present in the PROV instance of interest).
+<p>A lowercase symbol <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-4-Frame" class="MathJax"><nobr><span id="MathJax-Span-54" class="math"><span style="display: inline-block; position: relative; width: 49px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-55" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-56" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-57" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-58" class="mi">y</span><span style="font-family: STIXGeneral;" id="MathJax-Span-59" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-60" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-61" class="mo">.</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-62" class="mo">.</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.817em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-4" type="math/tex">x,y,...</script> on its own denotes an identifier.
+Identifiers may or may not be URIs.  Identifiers are viewed as variables in logic (or blank nodes in RDF): just because we have two different identifiers <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-5-Frame" class="MathJax"><nobr><span id="MathJax-Span-63" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-64" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-65" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-5" type="math/tex">x</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-6-Frame" class="MathJax"><nobr><span id="MathJax-Span-66" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.951em, -0.55em); top: -2.57em; left: 0em;"><span id="MathJax-Span-67" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-68" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.817em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-6" type="math/tex">y</script> doesn't tell us that they denote different things, since we could discover that they are actually the same later.  We write <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-7-Frame" class="MathJax"><nobr><span id="MathJax-Span-69" class="math"><span style="display: inline-block; position: relative; width: 75px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-70" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-71" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-72" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-73" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-74" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-75" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-76" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-77" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-78" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-79" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-80" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-81" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-7" type="math/tex">Identifiers</script> for the set of identifiers of interest in a given situation (typically, the set of identifiers present in the PROV instance of interest).
 </p>
 </section>
 
 <section id="times-and-intervals">
 <h3><span class="secno">2.2 </span> Times and Intervals </h3>
 
-<p>We assume a linearly ordered set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-8-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-82"><span style="display: inline-block; position: relative; width: 82px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.9px 19360px 54.4px -8.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-83"><span class="mo" id="MathJax-Span-84" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-85" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-86" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-87" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-88" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-89" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-90" style="font-family: STIXGeneral-Regular;">,</span><span class="mo" id="MathJax-Span-91" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">≤</span><span class="mo" id="MathJax-Span-92" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.5px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-8">(Times,\leq)</script> of time instants.  For convenience we assume the order is total or linear order, corresponding to a linear timeline; however, PROV does not assume that time is linear and events could be partially ordered and not necessarily reconciled to a single global clock.  
+<p>We assume a linearly ordered set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-8-Frame" class="MathJax"><nobr><span id="MathJax-Span-82" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.719em, 1000em, 2.922em, -0.478em); top: -2.57em; left: 0em;"><span id="MathJax-Span-83" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-84" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-86" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-87" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-88" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-89" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-90" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-91" class="mo">≤</span><span style="font-family: STIXGeneral;" id="MathJax-Span-92" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.038em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-8" type="math/tex">(Times,\leq)</script> of time instants.  For convenience we assume the order is total or linear order, corresponding to a linear timeline; however, PROV does not assume that time is linear and events could be partially ordered and not necessarily reconciled to a single global clock.  
 </p>
-<p>We also consider a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-9-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-93"><span style="display: inline-block; position: relative; width: 68px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-94"><span class="mi" id="MathJax-Span-95" style="font-family: STIXGeneral-Italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-96" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-97" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-98" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-99" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-100" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-101" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-102" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-103" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-9">Intervals</script> of closed intervals of the form <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-10-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-104"><span style="display: inline-block; position: relative; width: 127px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.6px 19360px 54.7px -7.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-105"><span class="mo" id="MathJax-Span-106" style="font-family: STIXGeneral-Regular;">{</span><span class="mi" id="MathJax-Span-107" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-108" style="font-family: STIXGeneral-Regular; padding-left: 6px;">∣</span><span class="msubsup" id="MathJax-Span-109" style="padding-left: 6px;"><span style="display: inline-block; position: relative; width: 14.1px; height: 0px;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-110" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 5.7px;"><span class="mn" id="MathJax-Span-111" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-112" style="font-family: STIXGeneral-Regular; padding-left: 6px;">≤</span><span class="mi" id="MathJax-Span-113" style="font-family: STIXGeneral-Italic; padding-left: 6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-114" style="font-family: STIXGeneral-Regular; padding-left: 6px;">≤</span><span class="msubsup" id="MathJax-Span-115" style="padding-left: 6px;"><span style="display: inline-block; position: relative; width: 14.1px; height: 0px;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-116" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 5.7px;"><span class="mn" id="MathJax-Span-117" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-118" style="font-family: STIXGeneral-Regular;">}</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19px; vertical-align: -4.7px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-10">\{t \mid t_1 \leq t \leq t_2\}</script>.
+<p>We also consider a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-9-Frame" class="MathJax"><nobr><span id="MathJax-Span-93" class="math"><span style="display: inline-block; position: relative; width: 62px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-94" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-95" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-97" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-98" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-99" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-100" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-101" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-102" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-103" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-9" type="math/tex">Intervals</script> of closed intervals of the form <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-10-Frame" class="MathJax"><nobr><span id="MathJax-Span-104" class="math"><span style="display: inline-block; position: relative; width: 108px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.705em, 1000em, 2.934em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-105" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-106" class="mo">{</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-107" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-108" class="mo">∣</span><span style="padding-left: 0.313em;" id="MathJax-Span-109" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-110" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-111" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-112" class="mo">≤</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-113" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-114" class="mo">≤</span><span style="padding-left: 0.313em;" id="MathJax-Span-115" class="msubsup"><span style="display: inline-block; position: relative; width: 0.7em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.274em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-117" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-118" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.066em; vertical-align: -0.265em;"></span></span></nobr></span><script id="MathJax-Element-10" type="math/tex">\{t \mid t_1 \leq t \leq t_2\}</script>.
 </p>
 </section>
 
 <section id="attributes-and-values">
 <h3><span class="secno">2.3 </span> Attributes and Values </h3>
 
-<p>We assume a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-11-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-119"><span style="display: inline-block; position: relative; width: 76px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -10px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-120"><span class="mi" id="MathJax-Span-121" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-122" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-123" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-124" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-125" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-126" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-127" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-128" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-129" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-130" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.5px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-11">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-12-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-131"><span style="display: inline-block; position: relative; width: 54px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -7.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-132"><span class="mi" id="MathJax-Span-133" style="font-family: STIXGeneral-Italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-134" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-135" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-136" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-137" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-138" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-12">Values</script>
+<p>We assume a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-11-Frame" class="MathJax"><nobr><span id="MathJax-Span-119" class="math"><span style="display: inline-block; position: relative; width: 69px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.758em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-120" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-121" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-122" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-123" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-124" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-125" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-126" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-127" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-128" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-129" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-130" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.87em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-11" type="math/tex">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-12-Frame" class="MathJax"><nobr><span id="MathJax-Span-131" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.45em); top: -2.57em; left: 0em;"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-134" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-135" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-136" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-137" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-138" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-12" type="math/tex">Values</script>
 of possible values of attributes.  To allow for the fact that some
 attributes can have undefined or multiple values, we sometimes use the set
-<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-13-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-139"><span style="display: inline-block; position: relative; width: 71px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-140"><span class="mi" id="MathJax-Span-141" style="font-family: STIXGeneral-Italic;">P</span><span class="mo" id="MathJax-Span-142" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-143" style="font-family: STIXGeneral-Italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-144" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-145" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-146" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-147" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-148" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-13">P(Value)</script>, that is, the set of sets of values.
+<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-13-Frame" class="MathJax"><nobr><span id="MathJax-Span-139" class="math"><span style="display: inline-block; position: relative; width: 63px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-140" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-141" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-142" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-143" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-144" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-145" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-146" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-147" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-148" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-13" type="math/tex">P(Value)</script>, that is, the set of sets of values.
 </p> </section>
 
 <section id="formulas">
 <h3><span class="secno">2.4 </span>Atomic Formulas</h3>
 
-<p>The following atomic formulas correspond to the statements of PROV-DM.  We assume that definitions 1-4 of PROV-CONSTRAINTS have been applied in order to expand all optional parameters; thus, we use uniform notation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-14-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-149"><span style="display: inline-block; position: relative; width: 116px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -8.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-150"><span class="mi" id="MathJax-Span-151" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mo" id="MathJax-Span-152" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-153" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-154" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mo" id="MathJax-Span-155" style="font-family: STIXGeneral-Regular;">,</span><span class="msubsup" id="MathJax-Span-156" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 18.5px; height: 0px;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-157" style="font-family: STIXGeneral-Italic;">a</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 10px;"><span class="mn" id="MathJax-Span-158" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-159" style="font-family: STIXGeneral-Regular;">,</span><span class="mo" id="MathJax-Span-160" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">…</span><span class="mo" id="MathJax-Span-161" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">,</span><span class="msubsup" id="MathJax-Span-162" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 18.5px; height: 0px;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-163" style="font-family: STIXGeneral-Italic;">a</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 10px;"><span class="mi" id="MathJax-Span-164" style="font-size: 70.7%; font-family: STIXGeneral-Italic;">n</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-165" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-14">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-15-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-166"><span style="display: inline-block; position: relative; width: 117px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -8.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-167"><span class="mi" id="MathJax-Span-168" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mo" id="MathJax-Span-169" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-170" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-171" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mo" id="MathJax-Span-172" style="font-family: STIXGeneral-Regular;">;</span><span class="msubsup" id="MathJax-Span-173" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 18.5px; height: 0px;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-174" style="font-family: STIXGeneral-Italic;">a</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 10px;"><span class="mn" id="MathJax-Span-175" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-176" style="font-family: STIXGeneral-Regular;">,</span><span class="mo" id="MathJax-Span-177" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">…</span><span class="mo" id="MathJax-Span-178" style="font-family: STIXGeneral-Regular; padding-left: 3.6px;">,</span><span class="msubsup" id="MathJax-Span-179" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 18.5px; height: 0px;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-180" style="font-family: STIXGeneral-Italic;">a</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 10px;"><span class="mi" id="MathJax-Span-181" style="font-size: 70.7%; font-family: STIXGeneral-Italic;">n</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-182" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-15">r(id;a_1,\ldots,a_n)</script>.
+<p>The following atomic formulas correspond to the statements of PROV-DM.  We assume that definitions 1-4 of PROV-CONSTRAINTS have been applied in order to expand all optional parameters; thus, we use uniform notation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-14-Frame" class="MathJax"><nobr><span id="MathJax-Span-149" class="math"><span style="display: inline-block; position: relative; width: 106px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.481em); top: -2.57em; left: 0em;"><span id="MathJax-Span-150" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-151" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-152" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-153" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-154" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-155" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-156" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-157" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-158" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-159" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-160" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-161" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-162" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-163" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-164" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-165" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-14" type="math/tex">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-15-Frame" class="MathJax"><nobr><span id="MathJax-Span-166" class="math"><span style="display: inline-block; position: relative; width: 107px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.481em); top: -2.57em; left: 0em;"><span id="MathJax-Span-167" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-168" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-169" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-170" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-171" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-172" class="mo">;</span><span style="padding-left: 0.188em;" id="MathJax-Span-173" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-174" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-175" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-176" class="mo">,</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-177" class="mo">…</span><span style="font-family: STIXGeneral; padding-left: 0.188em;" id="MathJax-Span-178" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-179" class="msubsup"><span style="display: inline-block; position: relative; width: 0.951em; height: 0px;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-180" class="mi">a</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.953em; left: 0.526em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-181" class="mi">n</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-182" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-15" type="math/tex">r(id;a_1,\ldots,a_n)</script>.
 </p>
-<p>Each parameter is either an identifier, a constant (e.g. a time or other literal value in an attribute list), or a null symbol "-".  Null symbols can only appear in the specified arguments in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-16-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-183"><span style="display: inline-block; position: relative; width: 151px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-184"><span class="mi" id="MathJax-Span-185" style="font-family: STIXGeneral-Italic;">w</span><span class="mi" id="MathJax-Span-186" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-187" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-188" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-189" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-190" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-191" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-192" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-193" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-194" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-195" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-196" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-197" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mi" id="MathJax-Span-198" style="font-family: STIXGeneral-Italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.4px;"></span></span><span class="mi" id="MathJax-Span-199" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-200" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-201" style="font-family: STIXGeneral-Italic;">h</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-16">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-17-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-202"><span style="display: inline-block; position: relative; width: 135px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-203"><span class="mi" id="MathJax-Span-204" style="font-family: STIXGeneral-Italic;">w</span><span class="mi" id="MathJax-Span-205" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-206" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-207" style="font-family: STIXGeneral-Italic;">D</span><span class="mi" id="MathJax-Span-208" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-209" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-210" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-211" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-212" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-213" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mi" id="MathJax-Span-214" style="font-family: STIXGeneral-Italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.7px;"></span></span><span class="mi" id="MathJax-Span-215" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-216" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-217" style="font-family: STIXGeneral-Italic;">m</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-17">wasDerivedFrom</script>, as shown in the grammar below.
+<p>Each parameter is either an identifier, a constant (e.g. a time or other literal value in an attribute list), or a null symbol "-".  Null symbols can only appear in the specified arguments in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-16-Frame" class="MathJax"><nobr><span id="MathJax-Span-183" class="math"><span style="display: inline-block; position: relative; width: 135px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-184" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-185" class="mi">w</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-186" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-187" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-188" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-189" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-190" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-191" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-192" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-193" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-194" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-195" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-196" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-197" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-198" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-199" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-200" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-201" class="mi">h</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-16" type="math/tex">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-17-Frame" class="MathJax"><nobr><span id="MathJax-Span-202" class="math"><span style="display: inline-block; position: relative; width: 121px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.511em); top: -2.57em; left: 0em;"><span id="MathJax-Span-203" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-204" class="mi">w</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-205" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-206" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-207" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-208" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-209" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-210" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-211" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-212" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-213" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-214" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-215" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-216" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-217" class="mi">m</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-17" type="math/tex">wasDerivedFrom</script>, as shown in the grammar below.
 </p>
-<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-18-Frame" style=""><nobr><span class="math" id="MathJax-Span-218"><span style="display: inline-block; position: relative; width: 539px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(-389.4px 19360px 461.7px -5.8px); top: -41px; left: 0px;"><span class="mrow" id="MathJax-Span-219"><span class="mtable" id="MathJax-Span-220" style="padding-right: 3.2px; padding-left: 3.2px;"><span style="display: inline-block; position: relative; width: 532.6px; height: 0px;"><span style="position: absolute; clip: rect(429.9px 19360px 1274.7px -9px); top: -858px; left: 0px;"><span style="display: inline-block; position: relative; width: 143px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.7px); top: -459.9px; right: 0px;"><span class="mtd" id="MathJax-Span-221"><span class="mrow" id="MathJax-Span-222"><span class="mi" id="MathJax-Span-223" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-224" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-225" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-226" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-227" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-228" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-229" style="font-family: STIXGeneral-Regular;">_</span><span class="mi" id="MathJax-Span-230" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-231" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-232" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-233" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-234" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-235" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-236" style="font-family: STIXGeneral-Italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -462.9px; right: 0px;"><span class="mtd" id="MathJax-Span-259"><span class="mrow" id="MathJax-Span-260"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -437px; right: 0px;"><span class="mtd" id="MathJax-Span-284"><span class="mrow" id="MathJax-Span-285"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.4px); top: -382.1px; right: 0px;"><span class="mtd" id="MathJax-Span-310"><span class="mrow" id="MathJax-Span-311"><span class="mi" id="MathJax-Span-312" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-313" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-314" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-315" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-316" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-317" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-318" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-319" style="font-family: STIXGeneral-Regular;">_</span><span class="mi" id="MathJax-Span-320" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-321" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-322" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-323" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-324" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-325" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-326" style="font-family: STIXGeneral-Italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -385.1px; right: 0px;"><span class="mtd" id="MathJax-Span-350"><span class="mrow" id="MathJax-Span-351"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -359.2px; right: 0px;"><span class="mtd" id="MathJax-Span-383"><span class="mrow" id="MathJax-Span-384"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.1px); top: -304.3px; right: 0px;"><span class="mtd" id="MathJax-Span-407"><span class="mrow" id="MathJax-Span-408"><span class="mi" id="MathJax-Span-409" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-410" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-411" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-412" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-413" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-414" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-415" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-416" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-417" style="font-family: STIXGeneral-Regular;">_</span><span class="mi" id="MathJax-Span-418" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-419" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-420" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-421" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-422" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-423" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-424" style="font-family: STIXGeneral-Italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -307.3px; right: 0px;"><span class="mtd" id="MathJax-Span-462"><span class="mrow" id="MathJax-Span-463"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -282px; right: 0px;"><span class="mtd" id="MathJax-Span-491"><span class="mrow" id="MathJax-Span-492"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -256px; right: 0px;"><span class="mtd" id="MathJax-Span-532"><span class="mrow" id="MathJax-Span-533"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -230.1px; right: 0px;"><span class="mtd" id="MathJax-Span-573"><span class="mrow" id="MathJax-Span-574"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -204.2px; right: 0px;"><span class="mtd" id="MathJax-Span-612"><span class="mrow" id="MathJax-Span-613"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -178.3px; right: 0px;"><span class="mtd" id="MathJax-Span-658"><span class="mrow" id="MathJax-Span-659"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -152.4px; right: 0px;"><span class="mtd" id="MathJax-Span-703"><span class="mrow" id="MathJax-Span-704"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -126.4px; right: 0px;"><span class="mtd" id="MathJax-Span-742"><span class="mrow" id="MathJax-Span-743"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -99.8px; right: 0px;"><span class="mtd" id="MathJax-Span-790"><span class="mrow" id="MathJax-Span-791"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -73.9px; right: 0px;"><span class="mtd" id="MathJax-Span-830"><span class="mrow" id="MathJax-Span-831"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -48px; right: 0px;"><span class="mtd" id="MathJax-Span-879"><span class="mrow" id="MathJax-Span-880"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -22.6px; right: 0px;"><span class="mtd" id="MathJax-Span-926"><span class="mrow" id="MathJax-Span-927"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 3.3px; right: 0px;"><span class="mtd" id="MathJax-Span-964"><span class="mrow" id="MathJax-Span-965"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 29.3px; right: 0px;"><span class="mtd" id="MathJax-Span-993"><span class="mrow" id="MathJax-Span-994"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 55.2px; right: 0px;"><span class="mtd" id="MathJax-Span-1027"><span class="mrow" id="MathJax-Span-1028"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.7px); top: 109.6px; right: 0px;"><span class="mtd" id="MathJax-Span-1050"><span class="mrow" id="MathJax-Span-1051"><span class="mi" id="MathJax-Span-1052" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1053" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1054" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mi" id="MathJax-Span-1055" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1056" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1057" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1058" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1059" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1060" style="font-family: STIXGeneral-Italic;">y</span><span class="mi" id="MathJax-Span-1061" style="font-family: STIXGeneral-Regular;">_</span><span class="mi" id="MathJax-Span-1062" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1063" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1064" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1065" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1066" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1067" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1068" style="font-family: STIXGeneral-Italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 106.4px; right: 0px;"><span class="mtd" id="MathJax-Span-1079"><span class="mrow" id="MathJax-Span-1080"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 132.2px; right: 0px;"><span class="mtd" id="MathJax-Span-1091"><span class="mrow" id="MathJax-Span-1092"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 158.1px; right: 0px;"><span class="mtd" id="MathJax-Span-1103"><span class="mrow" id="MathJax-Span-1104"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 183.4px; right: 0px;"><span class="mtd" id="MathJax-Span-1122"><span class="mrow" id="MathJax-Span-1123"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(34.4px 19360px 51.3px -8.7px); top: 237.9px; right: 0px;"><span class="mtd" id="MathJax-Span-1143"><span class="mrow" id="MathJax-Span-1144"><span class="mi" id="MathJax-Span-1145" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1146" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1147" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1148" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1149" style="font-family: STIXGeneral-Italic;">s</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(34.4px 19360px 55px -8.3px); top: 262.3px; right: 0px;"><span class="mtd" id="MathJax-Span-1180"><span class="mrow" id="MathJax-Span-1181"><span class="mi" id="MathJax-Span-1182" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1183" style="font-family: STIXGeneral-Italic;">y</span></span></span><span style="display: inline-block; 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width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -330.2px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-385"><span class="mrow" id="MathJax-Span-386"><span class="texatom" id="MathJax-Span-387"><span class="mrow" id="MathJax-Span-388"><span class="mo" id="MathJax-Span-389" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(36.1px 19360px 51.2px -7.4px); top: -304.3px; left: 50%; margin-left: -12px;"><span class="mtd" id="MathJax-Span-425"><span class="mrow" id="MathJax-Span-426"><span class="texatom" id="MathJax-Span-427"><span class="mrow" id="MathJax-Span-428"><span class="mo" id="MathJax-Span-429" style="font-family: STIXGeneral-Regular;">:<span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral-Regular; 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clip: rect(31.9px 19360px 51.3px -7.7px); top: -175.2px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-614"><span class="mrow" id="MathJax-Span-615"><span class="texatom" id="MathJax-Span-616"><span class="mrow" id="MathJax-Span-617"><span class="mo" id="MathJax-Span-618" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -149.3px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-660"><span class="mrow" id="MathJax-Span-661"><span class="texatom" id="MathJax-Span-662"><span class="mrow" id="MathJax-Span-663"><span class="mo" id="MathJax-Span-664" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -123.4px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-705"><span class="mrow" id="MathJax-Span-706"><span class="texatom" id="MathJax-Span-707"><span class="mrow" id="MathJax-Span-708"><span class="mo" id="MathJax-Span-709" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -97.4px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-744"><span class="mrow" id="MathJax-Span-745"><span class="texatom" id="MathJax-Span-746"><span class="mrow" id="MathJax-Span-747"><span class="mo" id="MathJax-Span-748" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -70.8px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-792"><span class="mrow" id="MathJax-Span-793"><span class="texatom" id="MathJax-Span-794"><span class="mrow" id="MathJax-Span-795"><span class="mo" id="MathJax-Span-796" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -44.9px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-832"><span class="mrow" id="MathJax-Span-833"><span class="texatom" id="MathJax-Span-834"><span class="mrow" id="MathJax-Span-835"><span class="mo" id="MathJax-Span-836" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -19px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-881"><span class="mrow" id="MathJax-Span-882"><span class="texatom" id="MathJax-Span-883"><span class="mrow" id="MathJax-Span-884"><span class="mo" id="MathJax-Span-885" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 6.4px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-928"><span class="mrow" id="MathJax-Span-929"><span class="texatom" id="MathJax-Span-930"><span class="mrow" id="MathJax-Span-931"><span class="mo" id="MathJax-Span-932" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 32.3px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-966"><span class="mrow" id="MathJax-Span-967"><span class="texatom" id="MathJax-Span-968"><span class="mrow" id="MathJax-Span-969"><span class="mo" id="MathJax-Span-970" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 58.3px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-995"><span class="mrow" id="MathJax-Span-996"><span class="texatom" id="MathJax-Span-997"><span class="mrow" id="MathJax-Span-998"><span class="mo" id="MathJax-Span-999" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 84.2px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1029"><span class="mrow" id="MathJax-Span-1030"><span class="texatom" id="MathJax-Span-1031"><span class="mrow" id="MathJax-Span-1032"><span class="mo" id="MathJax-Span-1033" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(36.1px 19360px 51.2px -7.4px); top: 109.6px; left: 50%; margin-left: -12px;"><span class="mtd" id="MathJax-Span-1069"><span class="mrow" id="MathJax-Span-1070"><span class="texatom" id="MathJax-Span-1071"><span class="mrow" id="MathJax-Span-1072"><span class="mo" id="MathJax-Span-1073" style="font-family: STIXGeneral-Regular;">:<span style="font-family: STIXGeneral-Regular; 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width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 187.1px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1105"><span class="mrow" id="MathJax-Span-1106"><span class="texatom" id="MathJax-Span-1107"><span class="mrow" id="MathJax-Span-1108"><span class="mo" id="MathJax-Span-1109" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 212.4px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1124"><span class="mrow" id="MathJax-Span-1125"><span class="texatom" id="MathJax-Span-1126"><span class="mrow" id="MathJax-Span-1127"><span class="mo" id="MathJax-Span-1128" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(36.1px 19360px 51.2px -7.4px); top: 237.9px; left: 50%; margin-left: -12px;"><span class="mtd" id="MathJax-Span-1150"><span class="mrow" id="MathJax-Span-1151"><span class="mo" id="MathJax-Span-1152" style="font-family: STIXGeneral-Regular;">:<span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">=</span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(36.1px 19360px 51.2px -7.4px); top: 262.3px; left: 50%; margin-left: -12px;"><span class="mtd" id="MathJax-Span-1184"><span class="mrow" id="MathJax-Span-1185"><span class="texatom" id="MathJax-Span-1186"><span class="mrow" id="MathJax-Span-1187"><span class="mo" id="MathJax-Span-1188" style="font-family: STIXGeneral-Regular;">:<span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral-Regular; 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font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-988" style="font-family: STIXGeneral-Regular;">,</span><span class="msubsup" id="MathJax-Span-989" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 16.5px; height: 0px;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.4px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-990" style="font-family: STIXGeneral-Italic;">e</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -39.1px; left: 8px;"><span class="mn" id="MathJax-Span-991" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-992" style="font-family: STIXGeneral-Regular;">)</span></span></span><span style="display: inline-block; 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 \begin{array}{rcl}
   atomic\_formula & {::=}& element\_formula\\
           & | & relation\_formula\\
@@ -1201,8 +1201,8 @@
 <h3><span class="secno">2.5 </span>First-Order Formulas</h3>
 
 <p>We also consider the usual connectives and quantifiers of
-first-order logic [<cite><a class="bibref" href="#bib-Logic">Logic</a></cite>].
-<span class="MathJax_Preview"></span><div class="MathJax_Display" role="textbox" aria-readonly="true" style="text-align: center;"><span class="MathJax" id="MathJax-Element-19-Frame" style=""><nobr><span class="math" id="MathJax-Span-1281"><span style="display: inline-block; position: relative; width: 213px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(-78.5px 19360px 150.8px -5.8px); top: -41px; left: 0px;"><span class="mrow" id="MathJax-Span-1282"><span class="mtable" id="MathJax-Span-1283" style="padding-right: 3.2px; padding-left: 3.2px;"><span style="display: inline-block; position: relative; width: 206.6px; height: 0px;"><span style="position: absolute; clip: rect(118.8px 19360px 341.8px -9px); top: -236px; left: 0px;"><span style="display: inline-block; position: relative; width: 12px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -149px; right: 0px;"><span class="mtd" id="MathJax-Span-1284"><span class="mrow" id="MathJax-Span-1285"><span class="mi" id="MathJax-Span-1286" style="font-family: STIXGeneral-Italic;">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -152.2px; right: 0px;"><span class="mtd" id="MathJax-Span-1308"><span class="mrow" id="MathJax-Span-1309"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -129px; right: 0px;"><span class="mtd" id="MathJax-Span-1321"><span class="mrow" id="MathJax-Span-1322"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -105.7px; right: 0px;"><span class="mtd" id="MathJax-Span-1335"><span class="mrow" id="MathJax-Span-1336"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -79.8px; right: 0px;"><span class="mtd" id="MathJax-Span-1347"><span class="mrow" id="MathJax-Span-1348"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -53.2px; right: 0px;"><span class="mtd" id="MathJax-Span-1363"><span class="mrow" id="MathJax-Span-1364"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -26.7px; right: 0px;"><span class="mtd" id="MathJax-Span-1379"><span class="mrow" id="MathJax-Span-1380"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: -0.1px; right: 0px;"><span class="mtd" id="MathJax-Span-1395"><span class="mrow" id="MathJax-Span-1396"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span><span style="position: absolute; clip: rect(74px 19360px 80px -9px); top: 25.8px; right: 0px;"><span class="mtd" id="MathJax-Span-1408"><span class="mrow" id="MathJax-Span-1409"></span></span><span style="display: inline-block; width: 0px; height: 77px;"></span></span></span><span style="display: inline-block; width: 0px; height: 236px;"></span></span><span style="position: absolute; clip: rect(119.1px 19360px 338.1px -7.7px); top: -232px; left: 33.8px;"><span style="display: inline-block; position: relative; width: 24px; height: 0px;"><span style="position: absolute; clip: rect(36.1px 19360px 51.2px -7.4px); top: -149px; left: 50%; margin-left: -12px;"><span class="mtd" id="MathJax-Span-1287"><span class="mrow" id="MathJax-Span-1288"><span class="texatom" id="MathJax-Span-1289"><span class="mrow" id="MathJax-Span-1290"><span class="mo" id="MathJax-Span-1291" style="font-family: STIXGeneral-Regular;">:<span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral-Regular; font-style: normal; font-weight: normal;">=</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -123.2px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1310"><span class="mrow" id="MathJax-Span-1311"><span class="texatom" id="MathJax-Span-1312"><span class="mrow" id="MathJax-Span-1313"><span class="mo" id="MathJax-Span-1314" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -100px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1323"><span class="mrow" id="MathJax-Span-1324"><span class="texatom" id="MathJax-Span-1325"><span class="mrow" id="MathJax-Span-1326"><span class="mo" id="MathJax-Span-1327" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -76.7px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1337"><span class="mrow" id="MathJax-Span-1338"><span class="texatom" id="MathJax-Span-1339"><span class="mrow" id="MathJax-Span-1340"><span class="mo" id="MathJax-Span-1341" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -50.8px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1349"><span class="mrow" id="MathJax-Span-1350"><span class="texatom" id="MathJax-Span-1351"><span class="mrow" id="MathJax-Span-1352"><span class="mo" id="MathJax-Span-1353" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: -24.2px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1365"><span class="mrow" id="MathJax-Span-1366"><span class="texatom" id="MathJax-Span-1367"><span class="mrow" id="MathJax-Span-1368"><span class="mo" id="MathJax-Span-1369" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 2.3px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1381"><span class="mrow" id="MathJax-Span-1382"><span class="texatom" id="MathJax-Span-1383"><span class="mrow" id="MathJax-Span-1384"><span class="mo" id="MathJax-Span-1385" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 28.9px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1397"><span class="mrow" id="MathJax-Span-1398"><span class="texatom" id="MathJax-Span-1399"><span class="mrow" id="MathJax-Span-1400"><span class="mo" id="MathJax-Span-1401" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.9px 19360px 51.3px -7.7px); top: 54.8px; left: 50%; margin-left: -2.5px;"><span class="mtd" id="MathJax-Span-1410"><span class="mrow" id="MathJax-Span-1411"><span class="texatom" id="MathJax-Span-1412"><span class="mrow" id="MathJax-Span-1413"><span class="mo" id="MathJax-Span-1414" style="font-family: STIXGeneral-Regular;">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="display: inline-block; width: 0px; height: 232px;"></span></span><span style="position: absolute; clip: rect(122.8px 19360px 349.8px -9px); top: -240px; left: 79.6px;"><span style="display: inline-block; position: relative; width: 127px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.7px); top: -149px; left: 0px;"><span class="mtd" id="MathJax-Span-1292"><span class="mrow" id="MathJax-Span-1293"><span class="mi" id="MathJax-Span-1294" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1295" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1296" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1297" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1298" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1299" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1300" style="font-family: STIXGeneral-Regular;">_</span><span class="mi" id="MathJax-Span-1301" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1302" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1303" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1304" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1305" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1306" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1307" style="font-family: STIXGeneral-Italic;">a</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(32.4px 19360px 51.2px -7.9px); top: -123.2px; left: 0px;"><span class="mtd" id="MathJax-Span-1315"><span class="mrow" id="MathJax-Span-1316"><span class="mi" id="MathJax-Span-1317" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1318" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1319" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1320" style="font-family: STIXGeneral-Italic;">e</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.8px); top: -100px; left: 0px;"><span class="mtd" id="MathJax-Span-1328"><span class="mrow" id="MathJax-Span-1329"><span class="mi" id="MathJax-Span-1330" style="font-family: STIXGeneral-Italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.7px;"></span></span><span class="mi" id="MathJax-Span-1331" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1332" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1333" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-1334" style="font-family: STIXGeneral-Italic;">e</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.1px); top: -76.7px; left: 0px;"><span class="mtd" id="MathJax-Span-1342"><span class="mrow" id="MathJax-Span-1343"><span class="mi" id="MathJax-Span-1344" style="font-family: STIXGeneral-Regular;">¬</span><span class="mtext" id="MathJax-Span-1345" style="font-family: STIXGeneral-Regular;">&nbsp;</span><span class="mi" id="MathJax-Span-1346" style="font-family: STIXGeneral-Italic;">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55.7px -8.5px); top: -50.8px; left: 0px;"><span class="mtd" id="MathJax-Span-1354"><span class="mrow" id="MathJax-Span-1355"><span class="msubsup" id="MathJax-Span-1356"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1357" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1358" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-1359" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∧</span><span class="msubsup" id="MathJax-Span-1360" style="padding-left: 4.8px;"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1361" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1362" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55.7px -8.5px); top: -24.2px; left: 0px;"><span class="mtd" id="MathJax-Span-1370"><span class="mrow" id="MathJax-Span-1371"><span class="msubsup" id="MathJax-Span-1372"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1373" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1374" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-1375" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∨</span><span class="msubsup" id="MathJax-Span-1376" style="padding-left: 4.8px;"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1377" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1378" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(32.8px 19360px 56.7px -8.5px); top: 1.3px; left: 0px;"><span class="mtd" id="MathJax-Span-1386"><span class="mrow" id="MathJax-Span-1387"><span class="msubsup" id="MathJax-Span-1388"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1389" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1390" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">1</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-1391" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⇒</span><span class="msubsup" id="MathJax-Span-1392" style="padding-left: 6px;"><span style="display: inline-block; position: relative; width: 20.5px; height: 0px;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.5px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-1393" style="font-family: STIXGeneral-Italic;">ϕ</span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -37.3px; left: 12px;"><span class="mn" id="MathJax-Span-1394" style="font-size: 70.7%; font-family: STIXGeneral-Regular;">2</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 49px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -9px); top: 28.9px; left: 0px;"><span class="mtd" id="MathJax-Span-1402"><span class="mrow" id="MathJax-Span-1403"><span class="mi" id="MathJax-Span-1404" style="font-family: STIXGeneral-Regular;">∀</span><span class="mi" id="MathJax-Span-1405" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-1406" style="font-family: STIXGeneral-Regular;">.</span><span class="mi" id="MathJax-Span-1407" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.6px); top: 54.8px; left: 0px;"><span class="mtd" id="MathJax-Span-1415"><span class="mrow" id="MathJax-Span-1416"><span class="mi" id="MathJax-Span-1417" style="font-family: STIXGeneral-Regular;">∃</span><span class="mi" id="MathJax-Span-1418" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-1419" style="font-family: STIXGeneral-Regular;">.</span><span class="mi" id="MathJax-Span-1420" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="display: inline-block; width: 0px; height: 240px;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 41px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 225.3px; vertical-align: -107.8px;"></span></span></nobr></span></div><script type="math/tex; mode=display" id="MathJax-Element-19">
+first-order logic [<cite><a href="#bib-Logic" class="bibref">Logic</a></cite>].
+<span class="MathJax_Preview"></span><div style="text-align: center;" aria-readonly="true" role="textbox" class="MathJax_Display"><span style="" id="MathJax-Element-19-Frame" class="MathJax"><nobr><span id="MathJax-Span-1281" class="math"><span style="display: inline-block; position: relative; width: 190px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(-4.148em, 1000em, 7.737em, -0.359em); top: -2.044em; left: 0em;"><span id="MathJax-Span-1282" class="mrow"><span style="padding-right: 0.167em; padding-left: 0.167em;" id="MathJax-Span-1283" class="mtable"><span style="display: inline-block; position: relative; width: 10.778em; height: 0px;"><span style="position: absolute; clip: rect(6.133em, 1000em, 17.695em, -0.526em); top: -12.208em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.643em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -7.787em; right: 0em;"><span id="MathJax-Span-1284" class="mtd"><span id="MathJax-Span-1285" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1286" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -7.856em; right: 0em;"><span id="MathJax-Span-1308" class="mtd"><span id="MathJax-Span-1309" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -6.656em; right: 0em;"><span id="MathJax-Span-1321" class="mtd"><span id="MathJax-Span-1322" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -5.456em; right: 0em;"><span id="MathJax-Span-1335" class="mtd"><span id="MathJax-Span-1336" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -4.118em; right: 0em;"><span id="MathJax-Span-1347" class="mtd"><span id="MathJax-Span-1348" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -2.745em; right: 0em;"><span id="MathJax-Span-1363" class="mtd"><span id="MathJax-Span-1364" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: -1.371em; right: 0em;"><span id="MathJax-Span-1379" class="mtd"><span id="MathJax-Span-1380" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: 0.002em; right: 0em;"><span id="MathJax-Span-1395" class="mtd"><span id="MathJax-Span-1396" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span><span style="position: absolute; clip: rect(3.797em, 1000em, 4.147em, -0.526em); top: 1.34em; right: 0em;"><span id="MathJax-Span-1408" class="mtd"><span id="MathJax-Span-1409" class="mrow"></span></span><span style="display: inline-block; width: 0px; height: 3.972em;"></span></span></span><span style="display: inline-block; width: 0px; height: 12.208em;"></span></span><span style="position: absolute; clip: rect(6.123em, 1000em, 17.476em, -0.459em); top: -11.974em; left: 1.768em;"><span style="display: inline-block; position: relative; width: 1.285em; height: 0px;"><span style="position: absolute; clip: rect(1.936em, 1000em, 2.756em, -0.445em); top: -7.787em; left: 50%; margin-left: -0.643em;"><span id="MathJax-Span-1287" class="mtd"><span id="MathJax-Span-1288" class="mrow"><span id="MathJax-Span-1289" class="texatom"><span id="MathJax-Span-1290" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1291" class="mo">:<span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">:</span><span style="font-family: STIXGeneral; font-style: normal; font-weight: normal;">=</span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -6.454em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1310" class="mtd"><span id="MathJax-Span-1311" class="mrow"><span id="MathJax-Span-1312" class="texatom"><span id="MathJax-Span-1313" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1314" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -5.254em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1323" class="mtd"><span id="MathJax-Span-1324" class="mrow"><span id="MathJax-Span-1325" class="texatom"><span id="MathJax-Span-1326" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1327" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -4.054em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1337" class="mtd"><span id="MathJax-Span-1338" class="mrow"><span id="MathJax-Span-1339" class="texatom"><span id="MathJax-Span-1340" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1341" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -2.716em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1349" class="mtd"><span id="MathJax-Span-1350" class="mrow"><span id="MathJax-Span-1351" class="texatom"><span id="MathJax-Span-1352" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1353" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: -1.343em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1365" class="mtd"><span id="MathJax-Span-1366" class="mrow"><span id="MathJax-Span-1367" class="texatom"><span id="MathJax-Span-1368" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1369" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 0.031em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1381" class="mtd"><span id="MathJax-Span-1382" class="mrow"><span id="MathJax-Span-1383" class="texatom"><span id="MathJax-Span-1384" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1385" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 1.404em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1397" class="mtd"><span id="MathJax-Span-1398" class="mrow"><span id="MathJax-Span-1399" class="texatom"><span id="MathJax-Span-1400" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1401" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.719em, 1000em, 2.759em, -0.459em); top: 2.742em; left: 50%; margin-left: -0.146em;"><span id="MathJax-Span-1410" class="mtd"><span id="MathJax-Span-1411" class="mrow"><span id="MathJax-Span-1412" class="texatom"><span id="MathJax-Span-1413" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1414" class="mo">|</span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="display: inline-block; width: 0px; height: 11.974em;"></span></span><span style="position: absolute; clip: rect(6.366em, 1000em, 18.134em, -0.524em); top: -12.442em; left: 4.178em;"><span style="display: inline-block; position: relative; width: 6.6em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.509em); top: -7.787em; left: 0em;"><span id="MathJax-Span-1292" class="mtd"><span id="MathJax-Span-1293" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1294" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1295" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1296" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1297" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1298" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1299" class="mi">c</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1300" class="mi">_</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1301" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1302" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1303" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1304" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1305" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1306" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1307" class="mi">a</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.742em, 1000em, 2.756em, -0.467em); top: -6.454em; left: 0em;"><span id="MathJax-Span-1315" class="mtd"><span id="MathJax-Span-1316" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1317" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1318" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1319" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1320" class="mi">e</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.758em, -0.518em); top: -5.254em; left: 0em;"><span id="MathJax-Span-1328" class="mtd"><span id="MathJax-Span-1329" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1330" class="mi">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.034em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1331" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1332" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1333" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1334" class="mi">e</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.478em); top: -4.054em; left: 0em;"><span id="MathJax-Span-1342" class="mtd"><span id="MathJax-Span-1343" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1344" class="mi">¬</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1345" class="mtext">&nbsp;</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1346" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: -2.716em; left: 0em;"><span id="MathJax-Span-1354" class="mtd"><span id="MathJax-Span-1355" class="mrow"><span id="MathJax-Span-1356" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1357" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1358" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1359" class="mo">∧</span><span style="padding-left: 0.25em;" id="MathJax-Span-1360" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1361" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1362" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: -1.343em; left: 0em;"><span id="MathJax-Span-1370" class="mtd"><span id="MathJax-Span-1371" class="mrow"><span id="MathJax-Span-1372" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1373" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1374" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1375" class="mo">∨</span><span style="padding-left: 0.25em;" id="MathJax-Span-1376" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1377" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1378" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.986em, -0.499em); top: 0.031em; left: 0em;"><span id="MathJax-Span-1386" class="mtd"><span id="MathJax-Span-1387" class="mrow"><span id="MathJax-Span-1388" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1389" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1390" class="mn">1</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1391" class="mo">⇒</span><span style="padding-left: 0.313em;" id="MathJax-Span-1392" class="msubsup"><span style="display: inline-block; position: relative; width: 1.068em; height: 0px;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.499em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1393" class="mi">ϕ</span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -1.862em; left: 0.643em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-1394" class="mn">2</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.524em); top: 1.404em; left: 0em;"><span id="MathJax-Span-1402" class="mtd"><span id="MathJax-Span-1403" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1404" class="mi">∀</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1405" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1406" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1407" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; clip: rect(1.712em, 1000em, 2.95em, -0.453em); top: 2.742em; left: 0em;"><span id="MathJax-Span-1415" class="mtd"><span id="MathJax-Span-1416" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1417" class="mi">∃</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1418" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1419" class="mo">.</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1420" class="mi">ϕ</span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="display: inline-block; width: 0px; height: 12.442em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 12.466em; vertical-align: -5.966em;"></span></span></nobr></span></div><script id="MathJax-Element-19" type="math/tex; mode=display">
 \begin{array}{rcl}
   \phi &{::=}& atomic\_formula\\
 & | & True\\
@@ -1243,26 +1243,26 @@
 <div class="note"><div class="note-title"><span>Note</span></div><div class="">TODO: Containment of things / collections? (for hadMember).</div></div>
 <p><em>Things</em> are things in the world.  Each thing has a lifetime during which it exists and attributes whose values can change over time.
 </p>
-<p>To model this, a structure <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-20-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1421"><span style="display: inline-block; position: relative; width: 18px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.4px 19360px 51.3px -7.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1422"><span class="mi" id="MathJax-Span-1423" style="font-family: STIXGeneral-Italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.4px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-20">W</script> includes:
+<p>To model this, a structure <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-20-Frame" class="MathJax"><nobr><span id="MathJax-Span-1421" class="math"><span style="display: inline-block; position: relative; width: 16px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.455em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1422" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1423" class="mi">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.073em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-20" type="math/tex">W</script> includes:
 </p>
-<div class="component" id="things" data-count="1" data-title="Component 1 (things)"><div class="ruleTitle"><a class="internalDFN" href="#things">Component 1 (things)</a></div><ol>
-  <li> a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-21-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1424"><span style="display: inline-block; position: relative; width: 55px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1425"><span class="mi" id="MathJax-Span-1426" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1427" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1428" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1429" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1430" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1431" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-21">Things</script> of things</li>
-  <li> a function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-22-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1432"><span style="display: inline-block; position: relative; width: 230px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1433"><span class="mi" id="MathJax-Span-1434" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1435" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1436" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1437" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1438" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1439" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1440" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1441" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1442" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-1443" style="font-family: STIXGeneral-Italic; padding-left: 6px;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1444" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1445" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1446" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1447" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1448" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1449" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mi" id="MathJax-Span-1450" style="font-family: STIXGeneral-Italic; padding-left: 6px;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-1451" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1452" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1453" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1454" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1455" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1456" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1457" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1458" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-22">lifetime : Things \to Intervals</script> from objects to time intervals</li>
-  <li>a function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-23-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1459"><span style="display: inline-block; position: relative; width: 386px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1460"><span class="mi" id="MathJax-Span-1461" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1462" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1463" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1464" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1465" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1466" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-1467" style="font-family: STIXGeneral-Italic; padding-left: 6px;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1468" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1469" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1470" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1471" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1472" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1473" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">×</span><span class="mi" id="MathJax-Span-1474" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">A</span><span class="mi" id="MathJax-Span-1475" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1476" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1477" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1478" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1479" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1480" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1481" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1482" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1483" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1484" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">×</span><span class="mi" id="MathJax-Span-1485" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1486" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1487" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1488" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1489" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1490" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mi" id="MathJax-Span-1491" style="font-family: STIXGeneral-Italic; padding-left: 6px;">P</span><span class="mo" id="MathJax-Span-1492" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1493" style="font-family: STIXGeneral-Italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1494" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1495" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1496" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1497" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1498" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1499" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-23">value : Things \times Attributes \times Times \to P(Values)</script>
+<div data-title="Component 1 (things)" data-count="1" class="component" id="things"><div class="ruleTitle"><a href="#things" class="internalDFN">Component 1 (things)</a></div><ol>
+  <li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-21-Frame" class="MathJax"><nobr><span id="MathJax-Span-1424" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1425" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1426" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1427" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1428" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1429" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1430" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1431" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-21" type="math/tex">Things</script> of things</li>
+  <li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-22-Frame" class="MathJax"><nobr><span id="MathJax-Span-1432" class="math"><span style="display: inline-block; position: relative; width: 207px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1433" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1434" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1435" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1436" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1437" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1438" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1439" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1440" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1441" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1442" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1443" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1444" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1445" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1446" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1447" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1448" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1449" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1450" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1451" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1452" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1453" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1454" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1455" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1456" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1457" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1458" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-22" type="math/tex">lifetime : Things \to Intervals</script> from objects to time intervals</li>
+  <li>a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-23-Frame" class="MathJax"><nobr><span id="MathJax-Span-1459" class="math"><span style="display: inline-block; position: relative; width: 349px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1460" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1461" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1462" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1463" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1464" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1465" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1466" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1467" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1468" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1469" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1470" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1471" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1472" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1473" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1474" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1475" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1476" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1477" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1478" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1479" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1480" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1481" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1482" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1483" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1484" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1485" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1486" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1487" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1488" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1489" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1490" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1491" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1492" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1493" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1494" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1495" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1496" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1497" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1498" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1499" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-23" type="math/tex">value : Things \times Attributes \times Times \to P(Values)</script>
 </li>
 </ol>
 </div>
 <p>
-The range of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-24-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1500"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1501"><span class="mi" id="MathJax-Span-1502" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1503" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1504" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1505" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1506" style="font-family: STIXGeneral-Italic;">e</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-24">value</script> is the set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-25-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1507"><span style="display: inline-block; position: relative; width: 78px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1508"><span class="mi" id="MathJax-Span-1509" style="font-family: STIXGeneral-Italic;">P</span><span class="mo" id="MathJax-Span-1510" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1511" style="font-family: STIXGeneral-Italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1512" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1513" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1514" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1515" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1516" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1517" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-25">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-26-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1518"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1519"><span class="mi" id="MathJax-Span-1520" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1521" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1522" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1523" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1524" style="font-family: STIXGeneral-Italic;">e</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-26">value</script>
-is essentially a multi-valued that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-27-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1525"><span style="display: inline-block; position: relative; width: 128px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 54.4px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1526"><span class="mi" id="MathJax-Span-1527" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1528" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1529" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1530" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1531" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1532" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1533" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-1534" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1535" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1536" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1537" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1538" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1539" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-1540" style="font-family: STIXVariants; padding-left: 6px;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.5px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-27">value(x,a,t) =
-\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-28-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1541"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1542"><span class="mi" id="MathJax-Span-1543" style="font-family: STIXGeneral-Italic;">a</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-28">a</script> is undefined for <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-29-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1544"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -9.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1545"><span class="mi" id="MathJax-Span-1546" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-29">x</script> at time <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-30-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1547"><span style="display: inline-block; position: relative; width: 6px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1548"><span class="mi" id="MathJax-Span-1549" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 12.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-30">t</script>.</p>
+The range of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-24-Frame" class="MathJax"><nobr><span id="MathJax-Span-1500" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1501" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1502" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1503" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1504" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1505" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1506" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-24" type="math/tex">value</script> is the set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-25-Frame" class="MathJax"><nobr><span id="MathJax-Span-1507" class="math"><span style="display: inline-block; position: relative; width: 70px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1508" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1509" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1510" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1511" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1512" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1513" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1514" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1515" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1516" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1517" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-25" type="math/tex">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-26-Frame" class="MathJax"><nobr><span id="MathJax-Span-1518" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1519" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1520" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1521" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1522" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1523" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1524" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-26" type="math/tex">value</script>
+is essentially a multi-valued that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-27-Frame" class="MathJax"><nobr><span id="MathJax-Span-1525" class="math"><span style="display: inline-block; position: relative; width: 116px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1526" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1527" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1528" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1529" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1530" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1531" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1532" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1533" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1534" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1535" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1536" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1537" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1538" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1539" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1540" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-27" type="math/tex">value(x,a,t) =
+\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-28-Frame" class="MathJax"><nobr><span id="MathJax-Span-1541" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1542" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1543" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-28" type="math/tex">a</script> is undefined for <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-29-Frame" class="MathJax"><nobr><span id="MathJax-Span-1544" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.553em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1545" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1546" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-29" type="math/tex">x</script> at time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-30-Frame" class="MathJax"><nobr><span id="MathJax-Span-1547" class="math"><span style="display: inline-block; position: relative; width: 5px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1548" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1549" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.721em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-30" type="math/tex">t</script>.</p>
 
 <p>Note that this description does not say what the structure of a
-<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-31-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1550"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1551"><span class="mi" id="MathJax-Span-1552" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1553" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1554" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1555" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1556" style="font-family: STIXGeneral-Italic;">g</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-31">Thing</script> is, only how it may be described in terms of its time interval
+<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-31-Frame" class="MathJax"><nobr><span id="MathJax-Span-1550" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1551" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1552" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1553" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1554" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1555" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1556" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-31" type="math/tex">Thing</script> is, only how it may be described in terms of its time interval
 and attribute values.  An object could just be a record of fixed
 attribute values; it could be a bear; it could be the Royal Society;
-it could be a transcendental number like <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-32-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1557"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.7px 19360px 51.3px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1558"><span class="mi" id="MathJax-Span-1559" style="font-family: STIXGeneral-Italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.6px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-32">\pi</script>.  All that matters from
-our point of view is that we know how to map the <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-33-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1560"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1561"><span class="mi" id="MathJax-Span-1562" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1563" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1564" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1565" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1566" style="font-family: STIXGeneral-Italic;">g</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-33">Thing</script> to its time interval and attribute mapping.
+it could be a transcendental number like <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-32-Frame" class="MathJax"><nobr><span id="MathJax-Span-1557" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.967em, 1000em, 2.763em, -0.507em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1558" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1559" class="mi">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.032em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.602em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-32" type="math/tex">\pi</script>.  All that matters from
+our point of view is that we know how to map the <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-33-Frame" class="MathJax"><nobr><span id="MathJax-Span-1560" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1561" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1562" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1563" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1564" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1565" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1566" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-33" type="math/tex">Thing</script> to its time interval and attribute mapping.
 </p>
 
 
@@ -1280,19 +1280,19 @@
 To model this, a structure includes:
 </p>
 
-<div class="component" id="objects" data-count="2" data-title="Component 2 (objects)"><div class="ruleTitle"><a class="internalDFN" href="#objects">Component 2 (objects)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-34-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1567"><span style="display: inline-block; position: relative; width: 58px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.8px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1568"><span class="mi" id="MathJax-Span-1569" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1570" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1571" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1572" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1573" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1574" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1575" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-34">Objects</script> 
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-35-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1576"><span style="display: inline-block; position: relative; width: 233px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1577"><span class="mi" id="MathJax-Span-1578" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1579" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1580" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1581" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1582" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1583" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1584" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1585" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1586" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-1587" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-1588" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1589" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1590" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1591" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1592" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1593" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1594" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mi" id="MathJax-Span-1595" style="font-family: STIXGeneral-Italic; padding-left: 6px;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-1596" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1597" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1598" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1599" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1600" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1601" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1602" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1603" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-35">lifetime : Objects \to Intervals</script> from objects to time intervals
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-36-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1604"><span style="display: inline-block; position: relative; width: 322px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1605"><span class="mi" id="MathJax-Span-1606" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1607" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1608" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1609" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1610" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1611" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-1612" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-1613" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1614" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1615" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1616" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1617" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1618" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1619" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">×</span><span class="mi" id="MathJax-Span-1620" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">A</span><span class="mi" id="MathJax-Span-1621" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1622" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1623" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1624" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1625" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1626" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1627" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1628" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1629" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1630" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mi" id="MathJax-Span-1631" style="font-family: STIXGeneral-Italic; padding-left: 6px;">P</span><span class="mo" id="MathJax-Span-1632" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1633" style="font-family: STIXGeneral-Italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1634" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1635" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1636" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1637" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1638" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1639" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-36">value : Objects \times Attributes \to P(Values)</script>
+<div data-title="Component 2 (objects)" data-count="2" class="component" id="objects"><div class="ruleTitle"><a href="#objects" class="internalDFN">Component 2 (objects)</a></div>
+  <ol><li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-34-Frame" class="MathJax"><nobr><span id="MathJax-Span-1567" class="math"><span style="display: inline-block; position: relative; width: 53px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.466em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1568" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1569" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1570" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1571" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1572" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1573" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1574" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1575" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-34" type="math/tex">Objects</script> 
+</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-35-Frame" class="MathJax"><nobr><span id="MathJax-Span-1576" class="math"><span style="display: inline-block; position: relative; width: 212px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1577" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1578" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1579" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1580" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1581" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1582" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1583" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1584" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1585" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1586" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1587" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1588" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1589" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1590" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1591" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1592" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1593" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1594" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1595" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1596" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1597" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1598" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1599" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1600" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1601" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1602" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1603" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-35" type="math/tex">lifetime : Objects \to Intervals</script> from objects to time intervals
+</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-36-Frame" class="MathJax"><nobr><span id="MathJax-Span-1604" class="math"><span style="display: inline-block; position: relative; width: 291px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1605" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1606" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1607" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1608" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1609" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1610" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1611" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1612" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1613" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1614" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1615" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1616" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1617" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1618" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-1619" class="mo">×</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-1620" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1621" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1622" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1623" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1624" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1625" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1626" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1627" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1628" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1629" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1630" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1631" class="mi">P</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1632" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1633" class="mi">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1634" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1635" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1636" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1637" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1638" class="mi">s</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1639" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-36" type="math/tex">value : Objects \times Attributes \to P(Values)</script>
 </li></ol>
 </div>
 
-<p>Intuitively, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-37-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1640"><span style="display: inline-block; position: relative; width: 79px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1641"><span class="mi" id="MathJax-Span-1642" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1643" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1644" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1645" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1646" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1647" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1648" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1649" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1650" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1651" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1652" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-37">lifetime(e)</script> is the time interval during which object
-<span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-38-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1653"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.4px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1654"><span class="mi" id="MathJax-Span-1655" style="font-family: STIXGeneral-Italic;">e</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-38">e</script> exists.  The set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-39-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1656"><span style="display: inline-block; position: relative; width: 79px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1657"><span class="mi" id="MathJax-Span-1658" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1659" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1660" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1661" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1662" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1663" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1664" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1665" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1666" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1667" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-39">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-40-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1668"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1669"><span class="mi" id="MathJax-Span-1670" style="font-family: STIXGeneral-Italic;">a</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-40">a</script> during the object's lifetime.
+<p>Intuitively, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-37-Frame" class="MathJax"><nobr><span id="MathJax-Span-1640" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1641" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1642" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1643" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1644" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1645" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1646" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1647" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1648" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1649" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1650" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1651" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1652" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-37" type="math/tex">lifetime(e)</script> is the time interval during which object
+<span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-38-Frame" class="MathJax"><nobr><span id="MathJax-Span-1653" class="math"><span style="display: inline-block; position: relative; width: 8px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.495em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1654" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1655" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-38" type="math/tex">e</script> exists.  The set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-39-Frame" class="MathJax"><nobr><span id="MathJax-Span-1656" class="math"><span style="display: inline-block; position: relative; width: 72px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1657" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1658" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1659" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1660" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1661" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1662" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1663" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1664" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1665" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1666" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1667" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-39" type="math/tex">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-40-Frame" class="MathJax"><nobr><span id="MathJax-Span-1668" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1669" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1670" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-40" type="math/tex">a</script> during the object's lifetime.
 </p>
 
-<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-41-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1671"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1672"><span class="mi" id="MathJax-Span-1673" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1674" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1675" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1676" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1677" style="font-family: STIXGeneral-Italic;">e</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-41">value</script> is sets of values,
-making <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-42-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1678"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 51.3px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1679"><span class="mi" id="MathJax-Span-1680" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1681" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1682" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1683" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1684" style="font-family: STIXGeneral-Italic;">e</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.6px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-42">value</script> effectively a multivalued function.  It is also possible to have two different objects that are indistinguishable by their attributes and time intervals.  Objects are not things, and the sets of <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-43-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1685"><span style="display: inline-block; position: relative; width: 58px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.8px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1686"><span class="mi" id="MathJax-Span-1687" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1688" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1689" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1690" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1691" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1692" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1693" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-43">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-44-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1694"><span style="display: inline-block; position: relative; width: 55px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1695"><span class="mi" id="MathJax-Span-1696" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1697" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1698" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1699" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1700" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1701" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-44">Things</script> are disjoint; however, certain objects, namely entities, are linked to things.
+<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-41-Frame" class="MathJax"><nobr><span id="MathJax-Span-1671" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1672" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1673" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1674" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1675" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1676" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1677" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-41" type="math/tex">value</script> is sets of values,
+making <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-42-Frame" class="MathJax"><nobr><span id="MathJax-Span-1678" class="math"><span style="display: inline-block; position: relative; width: 37px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.763em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1679" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1680" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1681" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1682" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1683" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1684" class="mi">e</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.875em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-42" type="math/tex">value</script> effectively a multivalued function.  It is also possible to have two different objects that are indistinguishable by their attributes and time intervals.  Objects are not things, and the sets of <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-43-Frame" class="MathJax"><nobr><span id="MathJax-Span-1685" class="math"><span style="display: inline-block; position: relative; width: 53px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.466em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1686" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1687" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1688" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1689" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1690" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1691" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1692" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1693" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-43" type="math/tex">Objects</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-44-Frame" class="MathJax"><nobr><span id="MathJax-Span-1694" class="math"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1695" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1696" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1697" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1698" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1699" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1700" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1701" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-44" type="math/tex">Things</script> are disjoint; however, certain objects, namely entities, are linked to things.
 
 </p>
 <section id="entities-1">
@@ -1300,28 +1300,28 @@
 
 An <em>entity</em> is a kind of object that describes a time-slice of a thing, during which some of the thing's attributes are fixed. We assume:
 
-<div class="component" id="entities" data-count="3" data-title="Component 3 (entities)"><div class="ruleTitle"><a class="internalDFN" href="#entities">Component 3 (entities)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-45-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1702"><span style="display: inline-block; position: relative; width: 141px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1703"><span class="mi" id="MathJax-Span-1704" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1705" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1706" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1707" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1708" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1709" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1710" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1711" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1712" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-1713" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-1714" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1715" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1716" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1717" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1718" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1719" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-45">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-46-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1720"><span style="display: inline-block; position: relative; width: 69px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.1px 19360px 51.3px -10px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1721"><span class="mi" id="MathJax-Span-1722" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-1723" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-1724" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1725" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1726" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1727" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1728" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1729" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1730" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1731" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.3px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-46">Activities</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-47-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1732"><span style="display: inline-block; position: relative; width: 51px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.4px 19360px 51.3px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1733"><span class="mi" id="MathJax-Span-1734" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1735" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1736" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1737" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1738" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1739" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-47">Events</script> below.
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-48-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1740"><span style="display: inline-block; position: relative; width: 224px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1741"><span class="mi" id="MathJax-Span-1742" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1743" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1744" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1745" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1746" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1747" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1748" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mo" id="MathJax-Span-1749" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-1750" style="font-family: STIXGeneral-Italic; padding-left: 6px;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1751" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1752" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1753" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1754" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1755" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1756" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1757" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-1758" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mi" id="MathJax-Span-1759" style="font-family: STIXGeneral-Italic; padding-left: 6px;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1760" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1761" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1762" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1763" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1764" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-48">thingOf : Entities \to Things</script> that associates each <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-49-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1765"><span style="display: inline-block; position: relative; width: 46px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.3px 19360px 55px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1766"><span class="mi" id="MathJax-Span-1767" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-1768" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1769" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1770" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1771" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1772" style="font-family: STIXGeneral-Italic;">y</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-49">Entity</script> with a <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-50-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1773"><span style="display: inline-block; position: relative; width: 48px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -7.9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1774"><span class="mi" id="MathJax-Span-1775" style="font-family: STIXGeneral-Italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 1.5px;"></span></span><span class="mi" id="MathJax-Span-1776" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1777" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1778" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1779" style="font-family: STIXGeneral-Italic;">g</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-50">Thing</script>, such that for each <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-51-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1780"><span style="display: inline-block; position: relative; width: 124px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1781"><span class="mi" id="MathJax-Span-1782" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1783" style="font-family: STIXGeneral-Regular; padding-left: 6px;">∈</span><span class="mi" id="MathJax-Span-1784" style="font-family: STIXGeneral-Italic; padding-left: 6px;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1785" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1786" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1787" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1788" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1789" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1790" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1791" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1792" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1793" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1794" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1795" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mo" id="MathJax-Span-1796" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-51">t \in lifetime(obj)</script>, and for each attribute <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-52-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1797"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1798"><span class="mi" id="MathJax-Span-1799" style="font-family: STIXGeneral-Italic;">a</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-52">a</script> we have <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-53-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1800"><span style="display: inline-block; position: relative; width: 305px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1801"><span class="mi" id="MathJax-Span-1802" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1803" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1804" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1805" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1806" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1807" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1808" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1809" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1810" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mo" id="MathJax-Span-1811" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1812" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1813" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1814" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-1815" style="font-family: STIXGeneral-Italic; padding-left: 6px;">v</span><span class="mi" id="MathJax-Span-1816" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1817" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1818" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1819" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1820" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1821" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1822" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1823" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1824" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1825" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1826" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1827" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mo" id="MathJax-Span-1828" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1829" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-1830" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-1831" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mo" id="MathJax-Span-1832" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1833" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1834" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1835" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1836" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1837" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-53">value(obj,a) \subseteq value(thingOf(obj),a,t)</script>.
-</li><li> <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-54-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1838"><span style="display: inline-block; position: relative; width: 179px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1839"><span class="mi" id="MathJax-Span-1840" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1841" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1842" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1843" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1844" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1845" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1846" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1847" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1848" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1849" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1850" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1851" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-1852" style="font-family: STIXGeneral-Italic; padding-left: 6px;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1853" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1854" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1855" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1856" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1857" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1858" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1859" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1860" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1861" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1862" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-54">lifetime(e) \subseteq lifetime(t)</script>.
+<div data-title="Component 3 (entities)" data-count="3" class="component" id="entities"><div class="ruleTitle"><a href="#entities" class="internalDFN">Component 3 (entities)</a></div>
+  <ol><li> a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-45-Frame" class="MathJax"><nobr><span id="MathJax-Span-1702" class="math"><span style="display: inline-block; position: relative; width: 128px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1703" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1704" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1705" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1706" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1707" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1708" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1709" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1710" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1711" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1712" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1713" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1714" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1715" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1716" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1717" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1718" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1719" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-45" type="math/tex">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-46-Frame" class="MathJax"><nobr><span id="MathJax-Span-1720" class="math"><span style="display: inline-block; position: relative; width: 64px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.763em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1721" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1722" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1723" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1724" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1725" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1726" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1727" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1728" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1729" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1730" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1731" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.859em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-46" type="math/tex">Activities</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-47-Frame" class="MathJax"><nobr><span id="MathJax-Span-1732" class="math"><span style="display: inline-block; position: relative; width: 46px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1733" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1734" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1735" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1736" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1737" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1738" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1739" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-47" type="math/tex">Events</script> below.
+</li><li> a function <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-48-Frame" class="MathJax"><nobr><span id="MathJax-Span-1740" class="math"><span style="display: inline-block; position: relative; width: 199px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1741" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1742" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1743" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1744" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1745" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1746" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1747" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1748" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1749" class="mo">:</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1750" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1751" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1752" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1753" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1754" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1755" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1756" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1757" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1758" class="mo">→</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1759" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1760" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1761" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1762" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1763" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1764" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-48" type="math/tex">thingOf : Entities \to Things</script> that associates each <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-49-Frame" class="MathJax"><nobr><span id="MathJax-Span-1765" class="math"><span style="display: inline-block; position: relative; width: 42px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.951em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1766" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1767" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1768" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1769" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1770" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1771" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1772" class="mi">y</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-49" type="math/tex">Entity</script> with a <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-50-Frame" class="MathJax"><nobr><span id="MathJax-Span-1773" class="math"><span style="display: inline-block; position: relative; width: 41px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.467em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1774" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1775" class="mi">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.077em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1776" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1777" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1778" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1779" class="mi">g</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-50" type="math/tex">Thing</script>, such that for each <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-51-Frame" class="MathJax"><nobr><span id="MathJax-Span-1780" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1781" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1782" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1783" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1784" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1785" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1786" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1787" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1788" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1789" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1790" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1791" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1792" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1793" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1794" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1795" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1796" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-51" type="math/tex">t \in lifetime(obj)</script>, and for each attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-52-Frame" class="MathJax"><nobr><span id="MathJax-Span-1797" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1798" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1799" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-52" type="math/tex">a</script> we have <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-53-Frame" class="MathJax"><nobr><span id="MathJax-Span-1800" class="math"><span style="display: inline-block; position: relative; width: 274px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1801" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1802" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1803" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1804" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1805" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1806" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1807" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1808" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1809" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1810" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1811" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1812" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1813" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1814" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1815" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1816" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1817" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1818" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1819" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1820" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1821" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1822" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1823" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1824" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1825" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1826" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1827" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1828" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1829" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1830" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1831" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1832" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1833" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1834" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1835" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1836" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1837" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-53" type="math/tex">value(obj,a) \subseteq value(thingOf(obj),a,t)</script>.
+</li><li> <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-54-Frame" class="MathJax"><nobr><span id="MathJax-Span-1838" class="math"><span style="display: inline-block; position: relative; width: 166px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1839" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1840" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1841" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1842" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1843" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1844" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1845" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1846" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1847" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1848" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1849" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1850" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1851" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1852" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1853" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1854" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1855" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1856" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1857" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1858" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1859" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1860" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1861" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1862" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-54" type="math/tex">lifetime(e) \subseteq lifetime(t)</script>.
 </li></ol>
 </div>
 
 <div class="remark"><p> Although both entities and things can have
   undefined or multiple attribute values, their meaning is slightly
-  different: for a thing, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-55-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1863"><span style="display: inline-block; position: relative; width: 128px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 54.4px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1864"><span class="mi" id="MathJax-Span-1865" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1866" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1867" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1868" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1869" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1870" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1871" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-1872" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1873" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1874" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1875" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1876" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1877" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-1878" style="font-family: STIXVariants; padding-left: 6px;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.5px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-55">value(x,a,t) = \emptyset</script> means that the
-  attribute <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-56-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1879"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1880"><span class="mi" id="MathJax-Span-1881" style="font-family: STIXGeneral-Italic;">a</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-56">a</script> has no value at time <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-57-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1882"><span style="display: inline-block; position: relative; width: 6px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1883"><span class="mi" id="MathJax-Span-1884" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 12.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-57">t</script>, whereas for an entity,
-  <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-58-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1885"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 54.4px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1886"><span class="mi" id="MathJax-Span-1887" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1888" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1889" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1890" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1891" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1892" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1893" style="font-family: STIXGeneral-Italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.1px;"></span></span><span class="mo" id="MathJax-Span-1894" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1895" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1896" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1897" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-1898" style="font-family: STIXVariants; padding-left: 6px;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.5px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-58">value(x,a) = \emptyset</script> only means that the entity does not record
-  a fixed value for <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-59-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1899"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(36.5px 19360px 51.2px -8.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1900"><span class="mi" id="MathJax-Span-1901" style="font-family: STIXGeneral-Italic;">a</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 10.8px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-59">a</script> forthe associated thing.  This does not imply
-  that <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-60-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1902"><span style="display: inline-block; position: relative; width: 202px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1903"><span class="mi" id="MathJax-Span-1904" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1905" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1906" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1907" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1908" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1909" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1910" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1911" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1912" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1913" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1914" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1915" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1916" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mo" id="MathJax-Span-1917" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1918" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1919" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1920" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1921" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1922" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1923" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1924" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1925" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-1926" style="font-family: STIXVariants; padding-left: 6px;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 20.1px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-60">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-61-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1927"><span style="display: inline-block; position: relative; width: 107px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1928"><span class="mi" id="MathJax-Span-1929" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1930" style="font-family: STIXGeneral-Regular; padding-left: 6px;">∈</span><span class="mi" id="MathJax-Span-1931" style="font-family: STIXGeneral-Italic; padding-left: 6px;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1932" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1933" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1934" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1935" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1936" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1937" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1938" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1939" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1940" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1941" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-61">t \in lifetime(e)</script>.
+  different: for a thing, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-55-Frame" class="MathJax"><nobr><span id="MathJax-Span-1863" class="math"><span style="display: inline-block; position: relative; width: 116px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1864" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1865" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1866" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1867" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1868" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1869" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1870" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1871" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1872" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1873" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1874" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1875" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1876" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1877" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1878" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-55" type="math/tex">value(x,a,t) = \emptyset</script> means that the
+  attribute <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-56-Frame" class="MathJax"><nobr><span id="MathJax-Span-1879" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1880" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1881" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-56" type="math/tex">a</script> has no value at time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-57-Frame" class="MathJax"><nobr><span id="MathJax-Span-1882" class="math"><span style="display: inline-block; position: relative; width: 5px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1883" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1884" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.721em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-57" type="math/tex">t</script>, whereas for an entity,
+  <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-58-Frame" class="MathJax"><nobr><span id="MathJax-Span-1885" class="math"><span style="display: inline-block; position: relative; width: 103px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1886" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1887" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1888" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1889" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1890" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1891" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1892" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1893" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1894" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1895" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1896" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1897" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1898" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.094em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-58" type="math/tex">value(x,a) = \emptyset</script> only means that the entity does not record
+  a fixed value for <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-59-Frame" class="MathJax"><nobr><span id="MathJax-Span-1899" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.954em, 1000em, 2.756em, -0.509em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1900" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1901" class="mi">a</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.609em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-59" type="math/tex">a</script> forthe associated thing.  This does not imply
+  that <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-60-Frame" class="MathJax"><nobr><span id="MathJax-Span-1902" class="math"><span style="display: inline-block; position: relative; width: 182px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1903" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1904" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1905" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1906" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1907" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1908" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1909" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1910" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1911" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1912" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1913" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1914" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1915" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1916" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1917" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1918" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1919" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1920" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1921" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1922" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1923" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1924" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1925" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-1926" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.127em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-60" type="math/tex">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-61-Frame" class="MathJax"><nobr><span id="MathJax-Span-1927" class="math"><span style="display: inline-block; position: relative; width: 100px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.488em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1928" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1929" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-1930" class="mo">∈</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-1931" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1932" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1933" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1934" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1935" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1936" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1937" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1938" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1939" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1940" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1941" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-61" type="math/tex">t \in lifetime(e)</script>.
 </p>
   <p>Furthermore, all of the attribute values of the entity must be
   present in the associated thing throughout the lifetime of the entity.
-For example, suppose <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-62-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1942"><span style="display: inline-block; position: relative; width: 167px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1943"><span class="mi" id="MathJax-Span-1944" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1945" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1946" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1947" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1948" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1949" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1950" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1951" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1952" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1953" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1954" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1955" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1956" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mo" id="MathJax-Span-1957" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1958" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1959" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-1960" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1961" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-1962" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-1963" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-1964" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-62">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-63-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1965"><span style="display: inline-block; position: relative; width: 26px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.5px -7.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1966"><span class="mo" id="MathJax-Span-1967" style="font-family: STIXGeneral-Regular;">{</span><span class="mn" id="MathJax-Span-1968" style="font-family: STIXGeneral-Regular;">1</span><span class="mo" id="MathJax-Span-1969" style="font-family: STIXGeneral-Regular;">}</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.7px; vertical-align: -4.5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-63">\{1\}</script> at some
-time in <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-64-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1970"><span style="display: inline-block; position: relative; width: 79px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1971"><span class="mi" id="MathJax-Span-1972" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1973" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1974" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mi" id="MathJax-Span-1975" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-1976" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1977" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1978" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-1979" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1980" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1981" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1982" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-64">lifetime(e)</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-65-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-1983"><span style="display: inline-block; position: relative; width: 226px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.2px 19360px 55px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-1984"><span class="mi" id="MathJax-Span-1985" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-1986" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-1987" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-1988" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-1989" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-1990" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1991" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-1992" style="font-family: STIXGeneral-Italic;">h</span><span class="mi" id="MathJax-Span-1993" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-1994" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-1995" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-1996" style="font-family: STIXGeneral-Italic;">O</span><span class="mi" id="MathJax-Span-1997" style="font-family: STIXGeneral-Italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 2.8px;"></span></span><span class="mo" id="MathJax-Span-1998" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-1999" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-2000" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-2001" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2002" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-2003" style="font-family: STIXGeneral-Regular;">,</span><span class="msup" id="MathJax-Span-2004" style="padding-left: 3.6px;"><span style="display: inline-block; position: relative; width: 13.5px; height: 0px;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-2005" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -49px; left: 7.1px;"><span class="mo" id="MathJax-Span-2006" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span><span class="mo" id="MathJax-Span-2007" style="font-family: STIXGeneral-Regular;">)</span><span class="mo" id="MathJax-Span-2008" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mo" id="MathJax-Span-2009" style="font-family: STIXGeneral-Regular; padding-left: 6px;">{</span><span class="mn" id="MathJax-Span-2010" style="font-family: STIXGeneral-Regular;">2</span><span class="mo" id="MathJax-Span-2011" style="font-family: STIXGeneral-Regular;">}</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 20.8px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-65">value(thingOf(e),a,t') = \{2\}</script> at some
-  other time <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-66-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2012"><span style="display: inline-block; position: relative; width: 14px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(23.2px 19360px 44.2px -8.3px); top: -41px; left: 0px;"><span class="mrow" id="MathJax-Span-2013"><span class="msup" id="MathJax-Span-2014"><span style="display: inline-block; position: relative; width: 13.5px; height: 0px;"><span style="position: absolute; clip: rect(34.4px 19360px 51.2px -8.3px); top: -48px; left: 0px;"><span class="mi" id="MathJax-Span-2015" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span><span style="position: absolute; top: -49px; left: 7.1px;"><span class="mo" id="MathJax-Span-2016" style="font-size: 70.7%; font-family: STIXVariants;">′</span><span style="display: inline-block; width: 0px; height: 42px;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 41px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 17px; vertical-align: -1.2px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-66">t'</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-67-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2017"><span style="display: inline-block; position: relative; width: 79px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.4px -8.6px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2018"><span class="mi" id="MathJax-Span-2019" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2020" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2021" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2022" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-2023" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-2024" style="font-family: STIXGeneral-Regular;">(</span><span class="mi" id="MathJax-Span-2025" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-2026" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2027" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">a</span><span class="mo" id="MathJax-Span-2028" style="font-family: STIXGeneral-Regular;">)</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.6px; vertical-align: -4.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-67">value(e,a)</script> must be <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-68-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2029"><span style="display: inline-block; position: relative; width: 10px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 52.4px -8.5px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2030"><span class="mi" id="MathJax-Span-2031" style="font-family: STIXVariants;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 17.5px; vertical-align: -2.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-68">\emptyset</script> because
+For example, suppose <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-62-Frame" class="MathJax"><nobr><span id="MathJax-Span-1942" class="math"><span style="display: inline-block; position: relative; width: 151px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1943" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1944" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1945" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1946" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1947" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1948" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1949" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1950" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1951" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1952" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1953" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1954" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1955" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1956" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1957" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1958" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1959" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1960" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1961" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1962" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-1963" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1964" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-62" type="math/tex">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-63-Frame" class="MathJax"><nobr><span id="MathJax-Span-1965" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1966" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-1967" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1968" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1969" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-63" type="math/tex">\{1\}</script> at some
+time in <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-64-Frame" class="MathJax"><nobr><span id="MathJax-Span-1970" class="math"><span style="display: inline-block; position: relative; width: 73px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.485em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1971" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1972" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1973" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1974" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1975" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1976" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1977" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1978" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1979" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1980" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1981" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1982" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-64" type="math/tex">lifetime(e)</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-65-Frame" class="MathJax"><nobr><span id="MathJax-Span-1983" class="math"><span style="display: inline-block; position: relative; width: 202px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.632em, 1000em, 2.952em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-1984" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1985" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1986" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1987" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1988" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1989" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-1990" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1991" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1992" class="mi">h</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1993" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1994" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1995" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1996" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1997" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-1998" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-1999" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2000" class="mo">)</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2001" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2002" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2003" class="mo">,</span><span style="padding-left: 0.188em;" id="MathJax-Span-2004" class="msup"><span style="display: inline-block; position: relative; width: 0.656em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2005" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -2.466em; left: 0.347em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-2006" class="mo">′</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span><span style="font-family: STIXGeneral;" id="MathJax-Span-2007" class="mo">)</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2008" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2009" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2010" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2011" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.162em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-65" type="math/tex">value(thingOf(e),a,t') = \{2\}</script> at some
+  other time <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-66-Frame" class="MathJax"><nobr><span id="MathJax-Span-2012" class="math"><span style="display: inline-block; position: relative; width: 11px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.107em, 1000em, 2.231em, -0.488em); top: -2.044em; left: 0em;"><span id="MathJax-Span-2013" class="mrow"><span id="MathJax-Span-2014" class="msup"><span style="display: inline-block; position: relative; width: 0.656em; height: 0px;"><span style="position: absolute; clip: rect(1.849em, 1000em, 2.756em, -0.488em); top: -2.57em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2015" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span><span style="position: absolute; top: -2.466em; left: 0.347em;"><span style="font-size: 70.7%; font-family: STIXVariants;" id="MathJax-Span-2016" class="mo">′</span><span style="display: inline-block; width: 0px; height: 2.103em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.044em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.953em; vertical-align: -0.074em;"></span></span></nobr></span><script id="MathJax-Element-66" type="math/tex">t'</script>.  Then <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-67-Frame" class="MathJax"><nobr><span id="MathJax-Span-2017" class="math"><span style="display: inline-block; position: relative; width: 72px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.922em, -0.506em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2018" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2019" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2020" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2021" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2022" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2023" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2024" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2025" class="mi">e</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2026" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2027" class="mi">a</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2028" class="mo">)</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.045em; vertical-align: -0.252em;"></span></span></nobr></span><script id="MathJax-Element-67" type="math/tex">value(e,a)</script> must be <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-68-Frame" class="MathJax"><nobr><span id="MathJax-Span-2029" class="math"><span style="display: inline-block; position: relative; width: 9px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.819em, -0.498em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2030" class="mrow"><span style="font-family: STIXVariants;" id="MathJax-Span-2031" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.984em; vertical-align: -0.142em;"></span></span></nobr></span><script id="MathJax-Element-68" type="math/tex">\emptyset</script> because
   there is no other set of values that is simultaneously contained in
-  both <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-69-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2032"><span style="display: inline-block; position: relative; width: 26px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.5px -7.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2033"><span class="mo" id="MathJax-Span-2034" style="font-family: STIXGeneral-Regular;">{</span><span class="mn" id="MathJax-Span-2035" style="font-family: STIXGeneral-Regular;">1</span><span class="mo" id="MathJax-Span-2036" style="font-family: STIXGeneral-Regular;">}</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.7px; vertical-align: -4.5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-69">\{1\}</script> and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-70-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2037"><span style="display: inline-block; position: relative; width: 26px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 54.5px -7.1px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2038"><span class="mo" id="MathJax-Span-2039" style="font-family: STIXGeneral-Regular;">{</span><span class="mn" id="MathJax-Span-2040" style="font-family: STIXGeneral-Regular;">2</span><span class="mo" id="MathJax-Span-2041" style="font-family: STIXGeneral-Regular;">}</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 18.7px; vertical-align: -4.5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-70">\{2\}</script>.
+  both <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-69-Frame" class="MathJax"><nobr><span id="MathJax-Span-2032" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2033" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2034" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2035" class="mn">1</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2036" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-69" type="math/tex">\{1\}</script> and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-70-Frame" class="MathJax"><nobr><span id="MathJax-Span-2037" class="math"><span style="display: inline-block; position: relative; width: 22px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.715em, 1000em, 2.926em, -0.426em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2038" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-2039" class="mo">{</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2040" class="mn">2</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2041" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.046em; vertical-align: -0.256em;"></span></span></nobr></span><script id="MathJax-Element-70" type="math/tex">\{2\}</script>.
 </p>
   </div>
 
@@ -1329,8 +1329,8 @@
 <h5><span class="secno">3.2.1.1 </span> Plans </h5>
 <p>We identify a specific subset of the entities called
   <em>plans</em>:
-</p><div class="component" id="plans" data-count="4" data-title="Component 4 (plans)"><div class="ruleTitle"><a class="internalDFN" href="#plans">Component 4 (plans)</a></div>
- <p> A set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-71-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2042"><span style="display: inline-block; position: relative; width: 127px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 53px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2043"><span class="mi" id="MathJax-Span-2044" style="font-family: STIXGeneral-Italic;">P</span><span class="mi" id="MathJax-Span-2045" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2046" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2047" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2048" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2049" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-2050" style="font-family: STIXGeneral-Italic; padding-left: 6px;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2051" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2052" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2053" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2054" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2055" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2056" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2057" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 17.2px; vertical-align: -3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-71">Plans \subseteq Entities</script> of plans.</p>
+</p><div data-title="Component 4 (plans)" data-count="4" class="component" id="plans"><div class="ruleTitle"><a href="#plans" class="internalDFN">Component 4 (plans)</a></div>
+ <p> A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-71-Frame" class="MathJax"><nobr><span id="MathJax-Span-2042" class="math"><span style="display: inline-block; position: relative; width: 115px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.848em, -0.526em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2043" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2044" class="mi">P</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2045" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2046" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2047" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2048" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2049" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2050" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2051" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2052" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2053" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2054" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2055" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2056" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2057" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.966em; vertical-align: -0.173em;"></span></span></nobr></span><script id="MathJax-Element-71" type="math/tex">Plans \subseteq Entities</script> of plans.</p>
   </div>
 <p></p></section>
 
@@ -1342,9 +1342,9 @@
 
 <p>An <em>activity</em> is an object that encompasses a set of events.  We introduce
 </p>
-<div class="component" id="activities" data-count="5" data-title="Component 5 (activities)"><div class="ruleTitle"><a class="internalDFN" href="#activities">Component 5 (activities)</a></div>
-  <ol><li>A set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-72-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2058"><span style="display: inline-block; position: relative; width: 151px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -10px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2059"><span class="mi" id="MathJax-Span-2060" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-2061" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2062" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2063" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2064" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2065" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2066" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2067" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2068" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2069" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2070" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-2071" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-2072" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-2073" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2074" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2075" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2076" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2077" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-72">Activities \subseteq Objects</script> of activities.</li>
-  <li> Activities are disjoint from Entities: <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-73-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2078"><span style="display: inline-block; position: relative; width: 180px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 52.4px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2079"><span class="mi" id="MathJax-Span-2080" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2081" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2082" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2083" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2084" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2085" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2086" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2087" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2088" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∩</span><span class="mi" id="MathJax-Span-2089" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">A</span><span class="mi" id="MathJax-Span-2090" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2091" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2092" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2093" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2094" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2095" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2096" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2097" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2098" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2099" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-2100" style="font-family: STIXVariants; padding-left: 6px;">∅</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 17.5px; vertical-align: -2.4px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-73">Entities\cap Activities
+<div data-title="Component 5 (activities)" data-count="5" class="component" id="activities"><div class="ruleTitle"><a href="#activities" class="internalDFN">Component 5 (activities)</a></div>
+  <ol><li>A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-72-Frame" class="MathJax"><nobr><span id="MathJax-Span-2058" class="math"><span style="display: inline-block; position: relative; width: 139px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2059" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2060" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2061" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2062" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2063" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2064" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2065" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2066" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2067" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2068" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2069" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2070" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2071" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2072" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2073" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2074" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2075" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2076" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2077" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-72" type="math/tex">Activities \subseteq Objects</script> of activities.</li>
+  <li> Activities are disjoint from Entities: <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-73-Frame" class="MathJax"><nobr><span id="MathJax-Span-2078" class="math"><span style="display: inline-block; position: relative; width: 168px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.666em, 1000em, 2.819em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2079" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2080" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2081" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2082" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2083" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2084" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2085" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2086" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2087" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2088" class="mo">∩</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2089" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2090" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2091" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2092" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2093" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2094" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2095" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2096" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2097" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2098" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2099" class="mo">=</span><span style="font-family: STIXVariants; padding-left: 0.313em;" id="MathJax-Span-2100" class="mi">∅</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.984em; vertical-align: -0.142em;"></span></span></nobr></span><script id="MathJax-Element-73" type="math/tex">Entities\cap Activities
   = \emptyset</script>.</li>
   </ol>
 </div></section>
@@ -1358,8 +1358,8 @@
   agent cannot be both entity and activity because the sets of
   entities and activities are disjoint.  We introduce:
 </p>
-<div class="component" id="agents" data-count="6" data-title="Component 6 (agents)"><div class="ruleTitle"><a class="internalDFN" href="#agents">Component 6 (agents)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-74-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2101"><span style="display: inline-block; position: relative; width: 135px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -10px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2102"><span class="mi" id="MathJax-Span-2103" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-2104" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-2105" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2106" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2107" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2108" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2109" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-2110" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-2111" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-2112" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2113" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2114" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2115" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2116" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-74">Agents \subseteq Objects</script> of agents.</p>
+<div data-title="Component 6 (agents)" data-count="6" class="component" id="agents"><div class="ruleTitle"><a href="#agents" class="internalDFN">Component 6 (agents)</a></div>
+  <p>A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-74-Frame" class="MathJax"><nobr><span id="MathJax-Span-2101" class="math"><span style="display: inline-block; position: relative; width: 121px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2102" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2103" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2104" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2105" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2106" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2107" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2108" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2109" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2110" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2111" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2112" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2113" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2114" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2115" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2116" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-74" type="math/tex">Agents \subseteq Objects</script> of agents.</p>
   </div>
 </section>
 
@@ -1367,23 +1367,23 @@
 <section id="interactions-1">
 <h4><span class="secno">3.2.4 </span> Interactions </h4>
 
-<p>We consider a set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-75-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2117"><span style="display: inline-block; position: relative; width: 176px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2118"><span class="mi" id="MathJax-Span-2119" style="font-family: STIXGeneral-Italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-2120" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2121" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2122" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2123" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2124" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2125" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2126" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2127" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2128" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2129" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2130" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2131" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-2132" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-2133" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-2134" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2135" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2136" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2137" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2138" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-75">Interactions \subseteq Objects</script> which are split into
+<p>We consider a set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-75-Frame" class="MathJax"><nobr><span id="MathJax-Span-2117" class="math"><span style="display: inline-block; position: relative; width: 160px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2118" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2119" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2120" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2121" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2122" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2123" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2124" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2125" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2126" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2127" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2128" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2129" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2130" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2131" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2132" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2133" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2134" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2135" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2136" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2137" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2138" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-75" type="math/tex">Interactions \subseteq Objects</script> which are split into
   <em>Events</em> connecting entities and activities,
   <em>Associations</em> between agents and activities,
   <em>Communications</em> between pairs of activities,
   <em>Delegations</em> between pairs of agents, and <em>Derivations</em> that describe chains of generation and usage steps.  (The first two sets may overlap.)  Interactions are disjoint from entities, activities and agents.
 </p>
-<div class="component" id="interactions" data-count="7" data-title="Component 7 (interactions)"><div class="ruleTitle"><a class="internalDFN" href="#interactions">Component 7 (interactions)</a></div>
-  <ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-76-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2139"><span style="display: inline-block; position: relative; width: 733px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2140"><span class="mi" id="MathJax-Span-2141" style="font-family: STIXGeneral-Italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-2142" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2143" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2144" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2145" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2146" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2147" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2148" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2149" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2150" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2151" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2152" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2153" style="font-family: STIXGeneral-Regular; padding-left: 6px;">=</span><span class="mi" id="MathJax-Span-2154" style="font-family: STIXGeneral-Italic; padding-left: 6px;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2155" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2156" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2157" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2158" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2159" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2160" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∪</span><span class="mi" id="MathJax-Span-2161" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">A</span><span class="mi" id="MathJax-Span-2162" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2163" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2164" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2165" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2166" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2167" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2168" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2169" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2170" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2171" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2172" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2173" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∪</span><span class="mi" id="MathJax-Span-2174" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2175" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2176" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-2177" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-2178" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-2179" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2180" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2181" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2182" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2183" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2184" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2185" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2186" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2187" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2188" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∪</span><span class="mi" id="MathJax-Span-2189" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">D</span><span class="mi" id="MathJax-Span-2190" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2191" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2192" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2193" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-2194" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2195" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2196" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2197" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2198" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2199" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2200" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∪</span><span class="mi" id="MathJax-Span-2201" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;">D</span><span class="mi" id="MathJax-Span-2202" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2203" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2204" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2205" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2206" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2207" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2208" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2209" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2210" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2211" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2212" style="font-family: STIXGeneral-Regular; padding-left: 6px;">⊆</span><span class="mi" id="MathJax-Span-2213" style="font-family: STIXGeneral-Italic; padding-left: 6px;">O</span><span class="mi" id="MathJax-Span-2214" style="font-family: STIXGeneral-Italic;">b</span><span class="mi" id="MathJax-Span-2215" style="font-family: STIXGeneral-Italic;">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2216" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2217" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2218" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2219" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-76">Interactions = Events \cup Associations \cup
+<div data-title="Component 7 (interactions)" data-count="7" class="component" id="interactions"><div class="ruleTitle"><a href="#interactions" class="internalDFN">Component 7 (interactions)</a></div>
+  <ol><li> A set <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-76-Frame" class="MathJax"><nobr><span id="MathJax-Span-2139" class="math"><span style="display: inline-block; position: relative; width: 672px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.952em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2140" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2141" class="mi">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.051em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2142" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2143" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2144" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2145" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2146" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2147" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2148" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2149" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2150" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2151" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2152" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2153" class="mo">=</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2154" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2155" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2156" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2157" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2158" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2159" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2160" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2161" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2162" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2163" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2164" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2165" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2166" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2167" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2168" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2169" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2170" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2171" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2172" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2173" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2174" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2175" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2176" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2177" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2178" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2179" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2180" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2181" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2182" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2183" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2184" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2185" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2186" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2187" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2188" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2189" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2190" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2191" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2192" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2193" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2194" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2195" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2196" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2197" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2198" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2199" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-2200" class="mo">∪</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.25em;" id="MathJax-Span-2201" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2202" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2203" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2204" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2205" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2206" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2207" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2208" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2209" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2210" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2211" class="mi">s</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-2212" class="mo">⊆</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-2213" class="mi">O</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2214" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2215" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2216" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2217" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2218" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2219" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.077em; vertical-align: -0.284em;"></span></span></nobr></span><script id="MathJax-Element-76" type="math/tex">Interactions = Events \cup Associations \cup
   Communications \cup Delegations \cup Derivations \subseteq Objects</script>
 </li>
-<li> A function <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-77-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2220"><span style="display: inline-block; position: relative; width: 1046px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -8.3px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2221"><span class="mi" id="MathJax-Span-2222" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2223" style="font-family: STIXGeneral-Italic;">y</span><span class="mi" id="MathJax-Span-2224" style="font-family: STIXGeneral-Italic;">p</span><span class="mi" id="MathJax-Span-2225" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-2226" style="font-family: STIXGeneral-Regular; padding-left: 6px;">:</span><span class="mi" id="MathJax-Span-2227" style="font-family: STIXGeneral-Italic; padding-left: 6px;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-2228" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2229" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2230" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2231" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2232" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2233" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2234" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2235" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2236" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2237" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2238" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2239" style="font-family: STIXGeneral-Regular; padding-left: 6px;">→</span><span class="mo" id="MathJax-Span-2240" style="font-family: STIXGeneral-Regular; padding-left: 6px;">{</span><span class="mi" id="MathJax-Span-2241" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2242" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2243" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2244" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2245" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mo" id="MathJax-Span-2246" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2247" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">e</span><span class="mi" id="MathJax-Span-2248" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2249" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mo" id="MathJax-Span-2250" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2251" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">u</span><span class="mi" id="MathJax-Span-2252" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2253" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2254" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-2255" style="font-family: STIXGeneral-Italic;">e</span><span class="mo" id="MathJax-Span-2256" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2257" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">g</span><span class="mi" id="MathJax-Span-2258" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2259" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2260" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2261" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2262" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2263" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2264" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2265" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2266" style="font-family: STIXGeneral-Italic;">n</span><span class="mo" id="MathJax-Span-2267" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2268" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">i</span><span class="mi" id="MathJax-Span-2269" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2270" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2271" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2272" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2273" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2274" style="font-family: STIXGeneral-Italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mi" id="MathJax-Span-2275" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2276" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2277" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2278" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2279" style="font-family: STIXGeneral-Italic;">n</span><span class="mo" id="MathJax-Span-2280" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2281" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.5px;"></span></span><span class="mi" id="MathJax-Span-2282" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2283" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2284" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2285" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2286" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2287" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2288" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2289" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2290" style="font-family: STIXGeneral-Italic;">n</span><span class="mo" id="MathJax-Span-2291" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2292" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">v</span><span class="mi" id="MathJax-Span-2293" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2294" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2295" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2296" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2297" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2298" style="font-family: STIXGeneral-Italic;">n</span><span class="mo" id="MathJax-Span-2299" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2300" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">q</span><span class="mi" id="MathJax-Span-2301" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-2302" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2303" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2304" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2305" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2306" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2307" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2308" style="font-family: STIXGeneral-Italic;">n</span><span class="mo" id="MathJax-Span-2309" style="font-family: STIXGeneral-Regular;">,</span><span class="mi" id="MathJax-Span-2310" style="font-family: STIXGeneral-Italic; padding-left: 3.6px;">p</span><span class="mi" id="MathJax-Span-2311" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2312" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2313" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-2314" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2315" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; 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height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2327" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2328" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2329" class="mi">b</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2330" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2331" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2332" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2333" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2334" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2335" class="mo">,</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.188em;" id="MathJax-Span-2336" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2337" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2338" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2339" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2340" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2341" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2342" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2343" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2344" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2345" class="mi">n</span><span style="font-family: STIXGeneral;" id="MathJax-Span-2346" class="mo">}</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.079em; vertical-align: -0.286em;"></span></span></nobr></span><script id="MathJax-Element-77" type="math/tex">type: Interactions \to
   \{start,end,usage,generation,invalidation,derivation,version,quotation,primarySource,attribution,delegation\}</script>.
 </li>
-<li> The sets <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-78-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2347"><span style="display: inline-block; position: relative; width: 51px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.4px 19360px 51.3px -9px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2348"><span class="mi" id="MathJax-Span-2349" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2350" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2351" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2352" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2353" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2354" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-78">Events</script>, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-79-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2355"><span style="display: inline-block; position: relative; width: 97px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.1px 19360px 51.3px -10px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2356"><span class="mi" id="MathJax-Span-2357" style="font-family: STIXGeneral-Italic;">A</span><span class="mi" id="MathJax-Span-2358" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2359" style="font-family: STIXGeneral-Italic;">s</span><span class="mi" id="MathJax-Span-2360" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2361" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2362" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2363" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2364" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2365" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2366" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2367" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2368" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.2px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-79">Associations</script>, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-80-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2369"><span style="display: inline-block; position: relative; width: 132px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.1px 19360px 51.3px -7.7px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2370"><span class="mi" id="MathJax-Span-2371" style="font-family: STIXGeneral-Italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2372" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2373" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-2374" style="font-family: STIXGeneral-Italic;">m</span><span class="mi" id="MathJax-Span-2375" style="font-family: STIXGeneral-Italic;">u</span><span class="mi" id="MathJax-Span-2376" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2377" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2378" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2379" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2380" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2381" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2382" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2383" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2384" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15.2px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-80">Communications</script>, <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-81-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2385"><span style="display: inline-block; position: relative; width: 93px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(31.8px 19360px 55px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2386"><span class="mi" id="MathJax-Span-2387" style="font-family: STIXGeneral-Italic;">D</span><span class="mi" id="MathJax-Span-2388" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2389" style="font-family: STIXGeneral-Italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0px;"></span></span><span class="mi" id="MathJax-Span-2390" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2391" style="font-family: STIXGeneral-Italic;">g</span><span class="mi" id="MathJax-Span-2392" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2393" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2394" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2395" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2396" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2397" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 19.2px; vertical-align: -5px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-81">Delegations</script>
-  and <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-82-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2398"><span style="display: inline-block; position: relative; width: 90px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(32.3px 19360px 51.3px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2399"><span class="mi" id="MathJax-Span-2400" style="font-family: STIXGeneral-Italic;">D</span><span class="mi" id="MathJax-Span-2401" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2402" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2403" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2404" style="font-family: STIXGeneral-Italic;">v</span><span class="mi" id="MathJax-Span-2405" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2406" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2407" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2408" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2409" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2410" style="font-family: STIXGeneral-Italic;">s</span></span><span style="display: inline-block; width: 0px; height: 48px;"></span></span></span><span style="border-left-width: 0px; border-left-style: solid; display: inline-block; overflow: hidden; width: 0px; height: 15px; vertical-align: -1.3px;"></span></span></nobr></span><script type="math/tex" id="MathJax-Element-82">Derivations</script> are all disjoint.
+<li> The sets <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-78-Frame" class="MathJax"><nobr><span id="MathJax-Span-2347" class="math"><span style="display: inline-block; position: relative; width: 46px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.742em, 1000em, 2.763em, -0.527em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2348" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2349" class="mi">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2350" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2351" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2352" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2353" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2354" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.843em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-78" type="math/tex">Events</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-79-Frame" class="MathJax"><nobr><span id="MathJax-Span-2355" class="math"><span style="display: inline-block; position: relative; width: 87px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.727em, 1000em, 2.758em, -0.577em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2356" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2357" class="mi">A</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2358" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2359" class="mi">s</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2360" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2361" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2362" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2363" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2364" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2365" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2366" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2367" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2368" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.854em; vertical-align: -0.076em;"></span></span></nobr></span><script id="MathJax-Element-79" type="math/tex">Associations</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-80-Frame" class="MathJax"><nobr><span id="MathJax-Span-2369" class="math"><span style="display: inline-block; position: relative; width: 117px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.729em, 1000em, 2.763em, -0.46em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2370" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2371" class="mi">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.022em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2372" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2373" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2374" class="mi">m</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2375" class="mi">u</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2376" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2377" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2378" class="mi">c</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2379" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2380" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2381" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2382" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2383" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2384" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.857em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-80" type="math/tex">Communications</script>, <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-81-Frame" class="MathJax"><nobr><span id="MathJax-Span-2385" class="math"><span style="display: inline-block; position: relative; width: 83px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.712em, 1000em, 2.951em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2386" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2387" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2388" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2389" class="mi">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2390" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2391" class="mi">g</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2392" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2393" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2394" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2395" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2396" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2397" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 1.076em; vertical-align: -0.283em;"></span></span></nobr></span><script id="MathJax-Element-81" type="math/tex">Delegations</script>
+  and <span class="MathJax_Preview"></span><span style="" aria-readonly="true" role="textbox" id="MathJax-Element-82-Frame" class="MathJax"><nobr><span id="MathJax-Span-2398" class="math"><span style="display: inline-block; position: relative; width: 81px; height: 0px; font-size: 107%;"><span style="position: absolute; clip: rect(1.741em, 1000em, 2.763em, -0.534em); top: -2.57em; left: 0em;"><span id="MathJax-Span-2399" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2400" class="mi">D</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2401" class="mi">e</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2402" class="mi">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.023em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2403" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2404" class="mi">v</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2405" class="mi">a</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2406" class="mi">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.018em;"></span></span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2407" class="mi">i</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2408" class="mi">o</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2409" class="mi">n</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-2410" class="mi">s</span></span><span style="display: inline-block; width: 0px; height: 2.57em;"></span></span></span><span style="border-left: 0em solid; display: inline-block; overflow: hidden; width: 0px; height: 0.844em; vertical-align: -0.082em;"></span></span></nobr></span><script id="MathJax-Element-82" type="math/tex">Derivations</script> are all disjoint.
 </li><li> Interactions are disjoint from entities, agents and
-activities:  <span class="MathJax_Preview"></span><span class="MathJax" id="MathJax-Element-83-Frame" role="textbox" aria-readonly="true" style=""><nobr><span class="math" id="MathJax-Span-2411"><span style="display: inline-block; position: relative; width: 373px; height: 0px; font-size: 121%;"><span style="position: absolute; clip: rect(30.9px 19360px 55px -9.2px); top: -48px; left: 0px;"><span class="mrow" id="MathJax-Span-2412"><span class="mi" id="MathJax-Span-2413" style="font-family: STIXGeneral-Italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 1px;"></span></span><span class="mi" id="MathJax-Span-2414" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2415" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2416" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2417" style="font-family: STIXGeneral-Italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2418" style="font-family: STIXGeneral-Italic;">a</span><span class="mi" id="MathJax-Span-2419" style="font-family: STIXGeneral-Italic;">c</span><span class="mi" id="MathJax-Span-2420" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2421" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2422" style="font-family: STIXGeneral-Italic;">o</span><span class="mi" id="MathJax-Span-2423" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2424" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2425" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∩</span><span class="mo" id="MathJax-Span-2426" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">(</span><span class="mi" id="MathJax-Span-2427" style="font-family: STIXGeneral-Italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.4px;"></span></span><span class="mi" id="MathJax-Span-2428" style="font-family: STIXGeneral-Italic;">n</span><span class="mi" id="MathJax-Span-2429" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2430" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2431" style="font-family: STIXGeneral-Italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.3px;"></span></span><span class="mi" id="MathJax-Span-2432" style="font-family: STIXGeneral-Italic;">i</span><span class="mi" id="MathJax-Span-2433" style="font-family: STIXGeneral-Italic;">e</span><span class="mi" id="MathJax-Span-2434" style="font-family: STIXGeneral-Italic;">s</span><span class="mo" id="MathJax-Span-2435" style="font-family: STIXGeneral-Regular; padding-left: 4.8px;">∪</span><span class="mi" id="MathJax-Span-2436" style="font-family: STIXGeneral-Italic; padding-left: 4.8px;&q